Quadratic equations
You also met quadratic equations last year. Here is a quick review, plus a very useful new tool: getting information from the solutions without solving the equation.
General form and formula
Apply the quadratic formula to find the roots of the following quadratic equation: $x^2 - 7x + 10 = 0$
Apply the quadratic formula to find the roots of the following quadratic equation:
$x^2 - 7x + 10 = 0$
Calculate the discriminant, $\Delta = b^2 - 4ac$, and state the number of real solutions the equation has: $4x^2 - 12x + 9 = 0$
Calculate the discriminant, $\Delta = b^2 - 4ac$, and state the number of real solutions the equation has:
$4x^2 - 12x + 9 = 0$
Incomplete equations
Solve the following incomplete quadratic equation by rearranging directly for the unknown: $3x^2 - 48 = 0$
Solve the following incomplete quadratic equation by rearranging directly for the unknown:
$3x^2 - 48 = 0$
Find the solutions to the following equation by taking out a common factor: $5x^2 + 15x = 0$
Find the solutions to the following equation by taking out a common factor:
$5x^2 + 15x = 0$
Solve the following equation by first eliminating the denominators to convert it into an equation with integer coefficients: $\frac{x^2}{2} - \frac{x}{3} = \frac{5}{6}$
Solve the following equation by first eliminating the denominators to convert it into an equation with integer coefficients:
$\frac{x^2}{2} - \frac{x}{3} = \frac{5}{6}$
The discriminant: how many solutions without solving
Write a quadratic equation in the form $x^2 + bx + c = 0$ knowing that its solutions are $x_1 = 4$ and $x_2 = -3$.
Write a quadratic equation in the form $x^2 + bx + c = 0$ knowing that its solutions are $x_1 = 4$ and $x_2 = -3$.
Discuss the number of real solutions to the equation $x^2 - (m+2)x + 9 = 0$ depending on the values of the parameter $m$.
Discuss the number of real solutions to the equation $x^2 - (m+2)x + 9 = 0$ depending on the values of the parameter $m$.
Cardano-Vieta: sum and product of the solutions
This property also works the other way round: to build an equation from its solutions.
In the equation $2x^2 + kx - 15 = 0$, one of the solutions is known to be $x = 3$. Calculate the value of the parameter $k$ and find the other solution.
In the equation $2x^2 + kx - 15 = 0$, one of the solutions is known to be $x = 3$. Calculate the value of the parameter $k$ and find the other solution.
Without solving the equation $3x^2 - 12x + c = 0$, find the value of the constant term $c$ knowing that the product of its roots is $3$. Also calculate the sum of these…
Without solving the equation $3x^2 - 12x + c = 0$, find the value of the constant term $c$ knowing that the product of its roots is $3$. Also calculate the sum of these roots using Vieta's formulas.
The perimeter of a rectangular plot is $68$ metres and its area is $240$ m$^2$. Use the relationship between the sum and product of the roots of a quadratic equation…
The perimeter of a rectangular plot is $68$ metres and its area is $240$ m$^2$. Use the relationship between the sum and product of the roots of a quadratic equation ($x^2 - Sx + P = 0$) to determine the dimensions of the plot.
Quadratic problems
The area of a rectangular solar panel is $24$ m$^2$. Knowing that its length is $2$ metres longer than its width $x$, calculate the exact dimensions of the panel.
The area of a rectangular solar panel is $24$ m$^2$. Knowing that its length is $2$ metres longer than its width $x$, calculate the exact dimensions of the panel.
A model rocket is launched vertically upwards. Its height $h$ in metres after $t$ seconds is given by the equation $h = 40t - 5t^2$. Determine at what times the rocket…
A model rocket is launched vertically upwards. Its height $h$ in metres after $t$ seconds is given by the equation $h = 40t - 5t^2$. Determine at what times the rocket will be at a height of exactly $75$ metres.