This document may contain unfilled definitions or examples. We will go over these during the class and the filled version will be posted after the lecture. Reveal all answers

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.