Integrated Math 2 / Unit 4: Solving Quadratic Equations
4.4 The Quadratic Formula
Notes
The quadratic formula solves any equation $ax^2 + bx + c = 0$: $\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
- Write the equation in standard form first and identify $a$, $b$ and $c$ with their signs.
- Example: $2x^2 + 3x - 5 = 0$ has $a = 2$, $b = 3$, $c = -5$. Then $\displaystyle x = \frac{-3 \pm \sqrt{9 + 40}}{4} = \frac{-3 \pm 7}{4}$, so $x = 1$ or $\displaystyle x = -\frac{5}{2}$.
- Simplify radicals in the answer when possible.
Practice
Solve $2x^2 + 3x - 5 = 0$ with the quadratic formula.
$b^2 - 4ac = 9 + 40 = 49$, so $\displaystyle x = \frac{-3 \pm 7}{4}$.
Solve $x^2 - 4x + 1 = 0$.
$\displaystyle x = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3}$.
To use the quadratic formula on $x^2 = 6x - 5$, you first move all terms to the left side. Which values of $a$, $b$, $c$ result?
Subtract $6x$ and add 5: $x^2 - 6x + 5 = 0$, so $b = -6$ and $c = 5$.
Ready for more? Take the Unit 4 test.