Quadratic equations
In 4th ESO you learned to solve $ax^2+bx+c=0$ using the quadratic formula. In Bachillerato this equation becomes a tool: you discuss how many solutions it has without solving it, you extract information from the coefficients without solving for anything, and you use it to reduce equations that look far more complicated. All the algebra that comes later relies on mastering this with ease.
General form and formula
Solve the following complete quadratic equation using the quadratic formula: $x^2 - 7x + 10 = 0$
Solve the following complete quadratic equation using the quadratic formula:
$x^2 - 7x + 10 = 0$
Incomplete equations: solving without the quadratic formula
Find the solutions to the following incomplete quadratic equation by factorising out a common factor: $3x^2 - 12x = 0$
Find the solutions to the following incomplete quadratic equation by factorising out a common factor:
$3x^2 - 12x = 0$
Solve the following incomplete quadratic equation by solving directly for the unknown: $2x^2 - 18 = 0$
Solve the following incomplete quadratic equation by solving directly for the unknown:
$2x^2 - 18 = 0$
The discriminant: the nature of the solutions
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 $k$ such that the equation $x^2 - kx + 9 = 0$ has a single repeated real solution.
Determine the value (or values) of the parameter $k$ such that the equation $x^2 - kx + 9 = 0$ has a single repeated real solution.
Vieta's formulas: sum and product without solving
Given the equation $4x^2 - 5x - 9 = 0$, calculate the sum and 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 product of its roots using Vieta's formulas, without needing to solve the equation.
Equations with a parameter: discussing and reducing to a quadratic
Two situations come up again and again in Bachillerato: an equation with an extra letter (a parameter, usually $m$ or $k$) for which you have to discuss the number of solutions; and a more complicated-looking equation that, with a change of variable, reduces to a quadratic.
In the parametric equation $2x^2 + mx + 15 = 0$, one of the solutions is known to be $x = 3$. Without using the quadratic formula, calculate the value of the parameter…
In the parametric equation $2x^2 + mx + 15 = 0$, one of the solutions is known to be $x = 3$. Without using the quadratic formula, calculate the value of the parameter $m$ and find the other solution.
Discuss the number of real solutions of 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 of the equation $(m-1)x^2 - 2mx + (m+2) = 0$ depending on the different values that the parameter $m$ can take.
Solve the following quartic equation by applying a suitable substitution to transform it into a quadratic equation: $(x^2 - 3x)^2 - 2(x^2 - 3x) - 8 = 0$
Solve the following quartic equation by applying a suitable substitution to transform it into a quadratic equation:
$(x^2 - 3x)^2 - 2(x^2 - 3x) - 8 = 0$
Quadratic problems
In a physics experiment, a water rocket is launched vertically upwards. Its height $h$ in metres 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 upwards. Its height $h$ in metres above the ground, as a function of time $t$ in seconds, is given by the expression $h(t) = -5t^2 + 20t + 2$. Determine the time or times at which the rocket is at a height of exactly $17$ metres.
A photograph measuring $20 \text{ cm}$ 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…
A photograph measuring $20 \text{ cm}$ 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.
In a right-angled triangle with legs $6 \text{ cm}$ and $8 \text{ cm}$, a rectangle is inscribed such that two of its sides lie along the legs and one of its vertices…
In a right-angled triangle with legs $6 \text{ cm}$ and $8 \text{ cm}$, a rectangle is inscribed such that two of its sides lie along 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.