# Lesson 16

The Quadratic Formula

- Let’s learn a formula for finding solutions to quadratic equations.

### Problem 1

For each equation, identify the values of \(a\), \(b\), and \(c\) that you would substitute into the quadratic formula to solve the equation.

- \(3x^2 + 8x + 4 = 0\)
- \(2x^2 - 5x + 2 = 0\)
- \(\text- 9x^2 + 13x - 1 = 0\)
- \(x^2 + x - 11 = 0\)
- \(\text- x^2 + 16x + 64 = 0\)

### Problem 2

Use the quadratic formula to show that the given solutions are correct.

- \(x^2 + 9x + 20 =0\). The solutions are \(x = \text- 4\) and \(x = \text- 5\).
- \(x^2 - 10x + 21 = 0\). The solutions are \(x = 3\) and \(x = 7\).
- \(3x^2 - 5x + 1 = 0\). The solutions are \(x = \frac56 \pm \frac{\sqrt{13}}{6}\).

### Problem 3

Select **all** the equations that are equivalent to \(81x^2+180x-200=100\)

\(81x^2+180x-100=0\)

\(81x^2+180x+100=200\)

\(81x^2+180x+100=400\)

\((9x+10)^2=400\)

\((9x+10)^2=0\)

\((9x-10)^2=10\)

\((9x-10)^2=20\)

### Problem 4

*Technology required.* Two objects are launched upward. Each function gives the distance from the ground in meters as a function of time, \(t\), in seconds.

Object A: \(f(t)=25+20t-5t^2\)

Object B: \(g(t)=30+10t-5t^2\)

Use graphing technology to graph each function.

- Which object reaches the ground first? Explain how you know.
- What is the maximum height of each object?

### Problem 5

Identify the values of \(a\), \(b\), and \(c\) that you would substitute into the quadratic formula to solve the equation.

- \(x^2 + 9x + 18 = 0\)
- \(4x^2 - 3x + 11 = 0\)
- \(81 - x + 5x^2 = 0\)
- \(\frac45 x^2 + 3x = \frac13 \)
- \(121 = x^2\)
- \(7x + 14x^2 = 42\)

### Problem 6

On the same coordinate plane, sketch a graph of each function.

- Function \(v\), defined by \(v(x) = |x+6|\)
- Function \(z\), defined by \(z(x)= |x|+9\)