Quadratic Equations

Quadratic Equations in Standard Form

Definition: Quadratic Equation in Standard Form

A quadratic equation in standard form is written as:

___________ $ax^{2} + bx + c = 0$ Reveal

where $a$, $b$, and $c$ are ___________________ real numbers Reveal and $a \neq$ ___________ $0$ Reveal.

Factoring Method

Definition: Zero-Product Property

If $ab = 0$ then ___________ $a = 0\text{ or }b = 0$ Reveal.

We use the Zero-Product Property to solve a quadratic equation by factoring.

1. Exercise

Solve: $3x^{2} - 2x - 5 = 0$ (Note: equation is in standard form)

$(3x - 5)(x + 1) = 0$
$3x - 5 = 0 \quad \text{or} \quad x + 1 = 0$
$x = \frac{5}{3} \quad \text{or} \quad x = -1$ Reveal

2. Exercise

Solve: $x = x^{2} - 12$

$x^2 - x - 12 = 0$
$(x-4)(x+3) = 0$
$x - 4 = 0 \quad \text{or} \quad x + 3 = 0$
$x = 4 \quad \text{or} \quad x = -3$ Reveal

Square Root Method

Use the square root method when $x^{2} = d$. Then $x =$ ___________ $\pm\sqrt{d}$ Reveal.

3. Exercise

Solve using the square root method: $2x^{2} - 1 = 17$

$2x^{2} = 18$
$x^{2} = 9$
$x = \pm\sqrt{9} = \pm 3$ Reveal

4. Exercise

Solve using the square root method: $(2x - 3)^{2} = 25$

$\sqrt{(2x - 3)^{2}} = \pm \sqrt{25}$
$2x - 3 = \pm 5$
$2x - 3 = 5 \quad \text{or} \quad 2x - 3 = -5$
$2x = 8 \quad \text{or} \quad 2x = -2$
$x = 4 \quad \text{or} \quad x = -1$ Reveal

Completing the Square

A perfect square trinomial can be written as the ___________________ the square of a binomial Reveal . For example:

• $x^{2} + 2ax + a^{2} = \underline{\blue{(x + a)^{2}}}$
• $x^{2} + 10x + 25 =$ ___________ $(x + 5)^{2}$ Reveal
• $x^{2} - 8x + 16 =$ ___________ $(x - 4)^{2}$ Reveal

5. Exercise:

Fill in the blank with the value that makes the trinomial into a perfect square trinomial.

• $x^{2} + 6x +$ ___________ $9$ Reveal $=$ ___________ $(x + 3)^{2}$ Reveal
• $x^{2} - 8x +$ ___________ $16$ Reveal $=$ ___________ $(x - 4)^{2}$ Reveal

Completing the Square Method

• Rearrange the quadratic equation so that __ $x^{2}$ Reveal and __ $x$ Reveal are on the left side of the equation and the ___________ constant Reveal is on the right side.
• Ensure the ___________ coefficient Reveal of $x^2$ is ___________ $1$ Reveal.
• Add the ___________ square Reveal of $\dfrac{1}{2}$ the ___________ coefficient Reveal of $x$ to both sides of the equation. i.e., $\left(\dfrac{b}{2}\right)^{2}$.
• Write the ___________ left side expression Reveal as the square of a binomial. Add the values on the right.
• Apply the ___________ Square Root Property Reveal.
• Solve for the ___________ variable Reveal.