Solving Quadratic Equations
Jayden launches a water balloon straight up from the roof of a 48-foot building. Its height after $t$ seconds is $h = -16t^2 + 32t + 48$. How long until it splashes on the ground? That is a quadratic equation, and you already know how to solve one. In Math III you go further. You will tell how many solutions an equation has without solving it, read information straight off the coefficients, and turn complicated-looking equations into quadratics.
The standard form and the quadratic formula
Where does the formula come from? It is what you get when you complete the square on $ax^2+bx+c=0$ and isolate $x$, so it works for every quadratic. If the trinomial factors easily, factoring is faster. If not, the formula always works.
Graphically, the solutions are the $x$-intercepts of the parabola $y=ax^2+bx+c$. Move the sliders and watch where it crosses the $x$-axis.
Many equations arrive in disguise. First multiply out, then collect everything on one side.
Solve the following complete quadratic equation by applying the quadratic formula: $x^2 - 7x + 10 = 0$
Solve the following complete quadratic equation by applying the quadratic formula:
$x^2 - 7x + 10 = 0$
Expand the expressions, simplify and solve the resulting quadratic equation: $(x - 2)(x + 3) = 2x - 6$
Expand the expressions, simplify and solve the resulting quadratic equation:
$(x - 2)(x + 3) = 2x - 6$
Incomplete quadratics: no formula needed
When $b=0$ or $c=0$ there are quicker routes than the formula.
If you get $x^2=-4$, there is no real solution, because no real number squared is negative. (In Math III you will also see that the complex solutions are $\pm 2i$.)
Find the solutions to the following incomplete quadratic equation by extracting a common factor: $3x^2 - 12x = 0$
Find the solutions to the following incomplete quadratic equation by extracting a common factor:
$3x^2 - 12x = 0$
Solve the following incomplete quadratic equation by directly isolating the unknown: $2x^2 - 18 = 0$
Solve the following incomplete quadratic equation by directly isolating the unknown:
$2x^2 - 18 = 0$
The discriminant: how many solutions?
The expression under the radical, $b^2-4ac$, decides how many real solutions there are before you compute anything.
The second example hints at a general fact. Look at what the signs of $a$ and $c$ do to $-4ac$.
Without solving the equation, determine the number of real solutions it has by calculating the value of its discriminant: $5x^2 - 3x + 2 = 0$
Without solving the equation, determine the number of real solutions it has by calculating the value of its discriminant:
$5x^2 - 3x + 2 = 0$
Determine the value (or values) of the parameter $k$ so that the equation $x^2 - kx + 9 = 0$ has a single double real solution.
Determine the value (or values) of the parameter $k$ so that the equation $x^2 - kx + 9 = 0$ has a single double real solution.
Prove algebraically that the quadratic equation $ax^2 + bx + c = 0$ (with $a \neq 0$) always has two distinct real solutions if the coefficients $a$ and $c$ have…
Prove algebraically that the quadratic equation $ax^2 + bx + c = 0$ (with $a \neq 0$) always has two distinct real solutions if the coefficients $a$ and $c$ have opposite signs (that is, if $a \cdot c < 0$ is satisfied).
Sum and product of the roots
They work in both directions. You can read the sum and product from an equation without solving it, or build an equation from its solutions:
$$x^2-(x_1+x_2)\,x+x_1x_2=0$$Given the equation $4x^2 - 5x - 9 = 0$, calculate the sum and the product of its roots using Vieta's formulas, without needing to solve the equation.
Given the equation $4x^2 - 5x - 9 = 0$, calculate the sum and the product of its roots using Vieta's formulas, without needing to solve the equation.
Write a quadratic equation in the form $ax^2 + bx + c = 0$ (with $a=1$) knowing that its solutions are $x_1 = 4$ and $x_2 = -3$.
Write a quadratic equation in the form $ax^2 + bx + c = 0$ (with $a=1$) knowing that its solutions are $x_1 = 4$ and $x_2 = -3$.
Let $x_1$ and $x_2$ be the roots of the equation $3x^2 - 5x + 1 = 0$. Without explicitly calculating the value of $x_1$ and $x_2$, use the properties of the roots to…
Let $x_1$ and $x_2$ be the roots of the equation $3x^2 - 5x + 1 = 0$. Without explicitly calculating the value of $x_1$ and $x_2$, use the properties of the roots to find the exact value of the expression $\frac{1}{x_1^2} + \frac{1}{x_2^2}$.
Roots and factored form
If $r_1$ and $r_2$ are the roots of $ax^2+bx+c$, then
$$ax^2+bx+c=a(x-r_1)(x-r_2)$$Each root gives a factor, and the leading coefficient $a$ stays in front. Move the sliders to see that the parabola $y=a(x-b)(x-c)$ always crosses the axis at $b$ and $c$.
If you know one root, plug it in. A root makes the equation true, so substitution gives you the unknown parameter.
Find the roots of the polynomial $P(x) = 2x^2 + 5x - 3$ and use them to write the factored polynomial.
Find the roots of the polynomial $P(x) = 2x^2 + 5x - 3$ and use them to write the factored polynomial.
In the parametric equation $2x^2 + mx + 15 = 0$, it is known that one of the solutions is $x = 3$. Without needing to use the quadratic formula, calculate the value of…
In the parametric equation $2x^2 + mx + 15 = 0$, it is known that one of the solutions is $x = 3$. Without needing to use the quadratic formula, calculate the value of the parameter $m$ and find the other solution.
Equations with a parameter
Sometimes the coefficients contain a letter, such as $k$. You are asked how many solutions there are for each value of that letter. First check whether the leading coefficient can be zero. If it can, the equation is not quadratic for that value. After that, study the sign of $D$.
In the widget below, the slider $a$ plays the role of $k$ in the example. Watch what happens at $a=0$ and at $a=3$.
Discuss the number of real solutions to the equation $(m-1)x^2 - 2mx + (m+2) = 0$ depending on the different values that the parameter $m$ can take.
Discuss the number of real solutions to the equation $(m-1)x^2 - 2mx + (m+2) = 0$ depending on the different values that the parameter $m$ can take.
Quadratic in form and rational equations
Many equations are not quadratic at first glance but become quadratic after a substitution, or after clearing denominators.
Rational equations work the same way once you clear the denominators. Multiply every term by the common denominator, and exclude any value that makes a denominator zero.
Solve the following fourth-degree equation by making a suitable change of variable to transform it into a quadratic equation: $(x^2 - 3x)^2 - 2(x^2 - 3x) - 8 = 0$
Solve the following fourth-degree equation by making a suitable change of variable to transform it into a quadratic equation:
$(x^2 - 3x)^2 - 2(x^2 - 3x) - 8 = 0$
Solve the following equation with algebraic fractions by working it out until it is reduced to a quadratic equation: $\frac{x+1}{x-2} + \frac{3x}{x+2} = 4$
Solve the following equation with algebraic fractions by working it out until it is reduced to a quadratic equation:
$\frac{x+1}{x-2} + \frac{3x}{x+2} = 4$
Modeling with quadratics: height and profit
When a quadratic function models a situation, a question about it becomes an equation. For height $=17$, set the function equal to $17$. For "hits the ground", set height $=0$. For "break-even", set profit $=0$. Then move everything to one side and solve. Finally, throw out any solution that makes no sense in context, such as a negative time.
Profit works the same way. If the profit in thousands of dollars is $P(x)=-x^2+10x-16$ for $x$ hundred items, break-even means $P(x)=0$. Then $x^2-10x+16=0$ gives $(x-2)(x-8)=0$, so the company breaks even at 200 and at 800 items.
In a physics experiment, a water rocket is launched vertically upward. Its height $h$ in feet above the ground, as a function of time $t$ in seconds, is given by the…
In a physics experiment, a water rocket is launched vertically upward. Its height $h$ in feet above the ground, as a function of time $t$ in seconds, is given by the expression $h(t) = -5t^2 + 20t + 2$. Determine at what instant or instants the rocket is exactly $17$ feet high.
A technology company manufactures external hard drives. The monthly profit, in thousands of dollars, when selling $x$ hundreds of units is modeled by the quadratic…
A technology company manufactures external hard drives. The monthly profit, in thousands of dollars, when selling $x$ hundreds of units is modeled by the quadratic function $B(x) = -2x^2 + 24x - 54$. How many units must be sold for the profit to be zero (break-even point)?
Area problems: set up and solve
A reliable plan: draw a picture, name the unknown, write the area in terms of it, set it equal to the given area, and solve.
An architect is designing a rectangular solar panel for the roof of an industrial building. Due to design constraints, the base of the panel must measure exactly $3$…
An architect is designing a rectangular solar panel for the roof of an industrial building. Due to design constraints, the base of the panel must measure exactly $3$ feet more than its height. If the total area of the panel must be $28 \text{ ft}^2$, set up a quadratic equation and calculate its dimensions.
A photograph $20 \text{ cm}$ high by $30 \text{ cm}$ wide is to be framed with a wooden frame of uniform width $x$. If the total area of the frame (excluding the…
A photograph $20 \text{ cm}$ high by $30 \text{ cm}$ wide is to be framed with a wooden frame of uniform width $x$. If the total area of the frame (excluding the photograph) is $216 \text{ cm}^2$, set up an equation and find the width of the frame.
Right-triangle problems
When a right angle appears, think of the Pythagorean Theorem, $a^2+b^2=c^2$. When a rectangle sits inside a triangle, think of similar triangles. Both lead to quadratics.
For a motion problem, remember that distance = rate $\times$ time. Two objects moving at right angles to each other are the legs of a right triangle, and the distance between them is the hypotenuse.
In a right triangle with legs $6 \text{ cm}$ and $8 \text{ cm}$, a rectangle is inscribed so that two of its sides lie on the legs and one of its vertices lies on the…
In a right triangle with legs $6 \text{ cm}$ and $8 \text{ cm}$, a rectangle is inscribed so that two of its sides lie on the legs and one of its vertices lies on the hypotenuse. If the area of the rectangle must be exactly $12 \text{ cm}^2$, set up a quadratic equation and find the possible dimensions of the rectangle.
Two surveillance drones depart simultaneously from the same base. The first drone flies in a straight line toward the north at a constant speed of $v \text{ km/h}$, and…
Two surveillance drones depart simultaneously from the same base. The first drone flies in a straight line toward the north at a constant speed of $v \text{ km/h}$, and the second flies toward the east at a speed of $v + 5 \text{ km/h}$. After $2$ hours of flight, the straight-line distance separating them is $50 \text{ km}$. Set up a quadratic equation and determine the speed of each drone.
Number, cost, and ratio problems
The product of two consecutive odd integers is $323$. Set up a quadratic equation and solve it to find what these two numbers are. Is there more than one possible pair?
The product of two consecutive odd integers is $323$. Set up a quadratic equation and solve it to find what these two numbers are. Is there more than one possible pair?
A group of friends decides to rent a sailboat for the weekend for a total cost of $\$360…
A group of friends decides to rent a sailboat for the weekend for a total cost of $\$360$, which they agree to pay in equal parts. At the last minute, two of the friends cannot attend, forcing the rest of the group to pay $\$9$ more each to cover the total cost. Set up an equation that reduces to a quadratic equation to determine how many friends made up the original group.
The golden ratio is obtained by dividing a segment of length $L$ into two parts $x$ and $y$ (with $x > y$) in such a way that the ratio of the total segment to the…
The golden ratio is obtained by dividing a segment of length $L$ into two parts $x$ and $y$ (with $x > y$) in such a way that the ratio of the total segment to the larger part is equal to the ratio of the larger part to the smaller part. If we set the length of the smaller part as $y = 1$, set up the resulting quadratic equation for $x$ and find the exact value of the golden ratio $\Phi$.