# Lesson 18

Solving Quadratics

- Let’s work to solve quadratic equations.

### 18.1: Math Talk: Operations with Roots

Evaluate mentally:

\(\sqrt{100}-15\)

\(\sqrt{125-10^2}\)

\(20-2\sqrt{49}\)

\(\sqrt{4^2+3^2}\)

### 18.2: Checking Brother’s Work

Priya's older brother is working on some higher-level math work and claims that \(x = 3\) is a solution to the equation \(x^3 - 5x^2 -2x = \text{-}24\).

- Explain how she could check that his solution is correct using each of these tools.
- A basic calculator
- A graphing tool

- When looking at his work, Priya sees that he has the equation \((x-3)(x^2 -2x - 8) = 0\). Knowing the zero product property holds, Priya recognizes that this equation means \(x-3 = 0\) or \(x^2 -2x - 8 = 0\) for this question. Find 2 other solutions to the original equation. Explain or show your reasoning.

### 18.3: Steps to Using the Quadratic Formula

The quadratic formula solves equations of the form \(ax^2 + bx + c = 0\) using the equation \(x=\frac{\text{-}b \pm \sqrt{b^2 - 4ac}}{2a}\).

Andre wants to use the quadratic formula to solve \(x^2 - 7x = \text{-}12\).

- What should Andre do first?
- What values of \(a, b,\) and \(c\) should he use?
- After substituting the values into the quadratic formula, what is the order he should use to calculate the solutions?
- Use the quadratic formula to solve the equation.
- Check your solutions.