Quadratic Equations

Solve $x^2 = 25$.

Solve $(x - 3)^2 = 0$.

Solve $(x - 2)(x - 5) = 0$.

Factorise $x^2 + 7x + 12$.

Factorise $x^2 - 7x + 12$.

Solve $x^2 + 5x + 6 = 0$.

Factorise $x^2 - 16$.

Solve $x^2 - 16 = 0$.

State the roots of $y = (x - 3)(x + 2)$.

Verify that $x = 3$ is a root of $x^2 - 5x + 6 = 0$.

Solve $x^2 - 3x - 10 = 0$.

Solve $x^2 = 5x + 6$.

Solve $x^2 = 5x$.

Expand $(x + 5)^2$.

Write $x^2 + 8x$ as $(x + p)^2 - q$.

Write $x^2 + 6x + 11$ in completed-square form $(x + p)^2 + q$.

Use the quadratic formula to solve $x^2 + 5x + 6 = 0$.

Solve $(x + 1)(x - 4) = 6$.

How many real roots does $x^2 - 4x + 4 = 0$ have?

Solve $(x + 3)^2 = 16$.

Factorise $2x^2 + 7x + 3$.

Use the quadratic formula to solve $2x^2 + 7x + 3 = 0$.

Write $x^2 - 6x + 4$ in completed-square form.

Solve $x^2 + 6x + 7 = 0$ by completing the square (give exact answers).

Solve $x^2 + 2x + 5 = 0$.

Solve $x^2 - 4x - 1 = 0$, giving exact answers.

The roots of $x^2 + 5x + 6 = 0$ are $\alpha$ and $\beta$. State $\alpha + \beta$ and $\alpha \beta$.

If $x = 3$ is a root of $x^2 + bx - 6 = 0$, find $b$.

Find a monic quadratic equation with roots $2$ and $5$.

Solve $(x - 4)^2 = 7$, giving exact answers.

For what values of $k$ does $x^2 + 6x + k = 0$ have two distinct real roots?

A rectangular garden is $x$ m wide and $(x + 3)$ m long. Its area is $40$ m². Find $x$.

Write $2x^2 - 8x + 5$ in the form $a(x + p)^2 + q$.

A ball is thrown upward and its height $h$ (m) after $t$ s is given by $h = 20t - 5t^2$. When does it hit the ground (other than at $t = 0$)?

Solve $2x^2 - 5x - 3 = 0$ exactly.

A right-angled triangle has legs of length $x$ and $(x + 7)$ cm. Its hypotenuse is $13$ cm. Find $x$.

The quadratic $x^2 - 8x + c = 0$ has roots that differ by 2. If one root is $5$, find $c$ and $b$.

Solve $x^2 + 6x + 4 = 0$ exactly, in surd form.

The equation $x^2 + kx + 9 = 0$ has exactly one (repeated) real root. Find $k$.

The product of two consecutive positive integers is $56$. Find the integers.