Quadratic Equations
A video game box has a square base with an area of 144 cm². How long is the side? You are looking for an $x$ such that $x^2 = 144$: this is a quadratic equation, because the unknown is squared. These equations can have two solutions, one, or none. There are two ways to solve them: factoring, when possible, and the quadratic formula, which always works.
General form
Graphically, $y = ax^2 + bx + c$ is a parabola, and the solutions are the points where it crosses the x-axis. Change $a$, $b$, and $c$ and see how many crossings there are:
For each of the following quadratic equations, identify the coefficients $a$, $b$ and $c$ and state whether it is a complete or incomplete equation: a)…
For each of the following quadratic equations, identify the coefficients $a$, $b$ and $c$ and state whether it is a complete or incomplete equation:
- a) $x^2 - 5x + 6 = 0$
- b) $2x^2 - 32 = 0$
- c) $x^2 + 8x = 0$
- d) $3 - x^2 + 2x = 0$
Check, without solving the equation, whether the proposed values of $x$ are solutions to the equation $x^2 - 4x + 3 = 0$: a) $x = 1$ b) $x = 0$ c) $x = 3$ d) $x = -1$
Check, without solving the equation, whether the proposed values of $x$ are solutions to the equation $x^2 - 4x + 3 = 0$:
- a) $x = 1$
- b) $x = 0$
- c) $x = 3$
- d) $x = -1$
Incomplete equations
They are solved without the formula, and faster:
Solve the following incomplete quadratic equations of the type $ax^2 + c = 0$. Remember that if the square root does not yield a real number, you must state so: a)…
Solve the following incomplete quadratic equations of the type $ax^2 + c = 0$. Remember that if the square root does not yield a real number, you must state so:
- a) $x^2 - 25 = 0$
- b) $3x^2 - 12 = 0$
- c) $x^2 + 16 = 0$
- d) $2x^2 = 50$
Solve the following incomplete quadratic equations of the type $ax^2 + bx = 0$ by taking out a common factor: a) $x^2 - 7x = 0$ b) $2x^2 + 10x = 0$ c) $3x^2 = 6x$ d)…
Solve the following incomplete quadratic equations of the type $ax^2 + bx = 0$ by taking out a common factor:
- a) $x^2 - 7x = 0$
- b) $2x^2 + 10x = 0$
- c) $3x^2 = 6x$
- d) $x^2 + x = 0$
The area of the square base of a video game storage box is $144\text{ cm}^2$. Calculate the length of the side of the base by setting up a quadratic equation.
The area of the square base of a video game storage box is $144\text{ cm}^2$. Calculate the length of the side of the base by setting up a quadratic equation.
Solving by factoring
The idea behind incomplete equations works for many more equations: if you can write $ax^2 + bx + c$ as a product of two factors, the equation practically solves itself. It is the first method required by the Secondary Year 3 curriculum, and it is usually the fastest.
1. Common factor
When there is no constant term ($c = 0$), every term has an $x$: take it out. $\ 5x^2 - 20x = 0 \Rightarrow 5x\,(x - 4) = 0 \Rightarrow x = 0$ or $x = 4$.
2. Difference of squares
One square minus another square factors as the product of two conjugate binomials:
$$x^2 - a^2 = (x + a)(x - a)$$So, $x^2 - 49 = 0 \Rightarrow (x + 7)(x - 7) = 0 \Rightarrow x = -7$ or $x = 7$. It also works with coefficients: $4x^2 - 9 = (2x + 3)(2x - 3) = 0 \Rightarrow x = \pm\frac{3}{2}$. Change $a$ and $b$ to see why $(a + b)(a - b) = a^2 - b^2$:
3. Trinomial $x^2 + bx + c$
It is the product of two binomials with a common term: $(x + p)(x + q) = x^2 + (p + q)\,x + p\,q$. Read backwards, to factor you look for two numbers $p$ and $q$:
The drawing shows $x^2 + 5x + 6 = (x + 2)(x + 3)$: the rectangle with sides $x + 2$ and $x + 3$ is split into a square $x^2$, two strips that add up to $5x$, and a rectangle of $2 \times 3 = 6$.
If you can't find the numbers (for example, in $x^2 - 5x - 1 = 0$ there are no integers that work), don't waste time: use the quadratic formula.
Solve the following incomplete quadratic equations using the fastest method (taking out the common factor or rearranging directly for the square root): a)…
Solve the following incomplete quadratic equations using the fastest method (taking out the common factor or rearranging directly for the square root):
- a) $4x^2 - 100 = 0$
- b) $3x^2 + 15x = 0$
- c) $\frac{x^2}{2} - 8 = 0$
- d) $5x^2 = 2x$
Given the quadratic equation $x^2 + kx + 12 = 0$, calculate the value of the coefficient $k$ knowing that one of the solutions to the equation is $x = 3$. Once $k$ is…
Given the quadratic equation $x^2 + kx + 12 = 0$, calculate the value of the coefficient $k$ knowing that one of the solutions to the equation is $x = 3$. Once $k$ is found, find the other solution of the equation.
The quadratic formula
For any complete equation, whether or not it can be factored:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$The $\pm$ gives two solutions: one by adding the root and another by subtracting it.
Apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions to the following complete equations: a) $x^2 - 6x + 8 = 0$ b)…
Apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions to the following complete equations:
- a) $x^2 - 6x + 8 = 0$
- b) $x^2 + x - 12 = 0$
Solve the following quadratic equation by first simplifying the algebraic expressions through expanding the algebraic identities: $$(2x - 3)^2 - (x + 1)^2 = -5$$…
Solve the following quadratic equation by first simplifying the algebraic expressions through expanding the algebraic identities:
$$(2x - 3)^2 - (x + 1)^2 = -5$$Remember that to apply the quadratic formula, you must first reduce the equation to the form $ax^2 + bx + c = 0$.
The discriminant: how many solutions?
What is inside the square root, $\Delta = b^2 - 4ac$, is called the discriminant and tells you how many solutions there are without solving:
| $\Delta = b^2 - 4ac$ | Solutions |
|---|---|
| $\Delta > 0$ | Two distinct solutions |
| $\Delta = 0$ | One solution (double): $x = -\frac{b}{2a}$ |
| $\Delta < 0$ | No real solution (the square root of a negative doesn't exist) |
Calculate the value of the discriminant $\Delta = b^2 - 4ac$ for each equation and state whether it has two real solutions, a repeated root, or no real solutions (you do…
Calculate the value of the discriminant $\Delta = b^2 - 4ac$ for each equation and state whether it has two real solutions, a repeated root, or no real solutions (you do not need to calculate the solutions):
- a) $x^2 - 4x + 4 = 0$
- b) $x^2 + 2x + 5 = 0$
- c) $x^2 - 5x - 1 = 0$
Study the discriminant ($\Delta = b^2 - 4ac$) of the following equations and determine, without solving them, how many real solutions each one has. Justify your answer…
Study the discriminant ($\Delta = b^2 - 4ac$) of the following equations and determine, without solving them, how many real solutions each one has. Justify your answer in each case:
- a) $x^2 - 6x + 9 = 0$
- b) $2x^2 + 3x + 5 = 0$
- c) $3x^2 - 4x - 2 = 0$
What value should the constant term take in equation a) so that it suddenly has no solution?
Given the quadratic equation $x^2 - (k-2)x + k = 0$, determine what values the parameter $k$ must take so that the equation has a single real solution (a repeated root).
Given the quadratic equation $x^2 - (k-2)x + k = 0$, determine what values the parameter $k$ must take so that the equation has a single real solution (a repeated root).
Extension: biquadratic equations
This type of equation is not part of the Secondary Year 3 curriculum, but it is useful for high school. In $x^4 - 13x^2 + 36 = 0$ only $x^4$ and $x^2$ appear. The trick is a change of variable: we call $t = x^2$ (so $x^4 = t^2$).
Biquadratic equations can be solved by using a substitution $t = x^2$. Use this technique to find the four real solutions of the following equation: $$x^4 - 13x^2 + 36 =…
Biquadratic equations can be solved by using a substitution $t = x^2$. Use this technique to find the four real solutions of the following equation:
$$x^4 - 13x^2 + 36 = 0$$Quadratic word problems
When solving a problem, discard the solutions that make no sense: a negative length, a number of people with decimals…
If we add three times an integer to its square, we get zero as a result. What numbers could these be? Set up the equation and find the possible solutions.
If we add three times an integer to its square, we get zero as a result. What numbers could these be? Set up the equation and find the possible solutions.
The product of two consecutive natural numbers is equal to their sum increased by $19$ units. What are these numbers?
The product of two consecutive natural numbers is equal to their sum increased by $19$ units. What are these numbers?
We want to put a wooden frame of constant width $x$ around a rectangular photograph measuring $10\text{ cm}$ by $15\text{ cm}$. If we know that the total area (photo…
We want to put a wooden frame of constant width $x$ around a rectangular photograph measuring $10\text{ cm}$ by $15\text{ cm}$. If we know that the total area (photo plus frame) must be $300\text{ cm}^2$, what should the width $x$ of the frame be?