Quadratic Formula
Use the quadratic formula to solve any quadratic equation in Grade 9 Algebra. Calculate the discriminant first to predict the number and type of solutions.
Key Concepts
Property For the quadratic equation $ax^2 + bx + c = 0$, $$ x = \frac{ b \pm \sqrt{b^2 4ac}}{2a} \quad \text{when } a \ne 0. $$ Explanation It's the ultimate 'cheat code' for solving any quadratic equation. Derived from completing the square, this formula saves you the trouble. Just identify your $a$, $b$, and $c$ values from the standard form of the equation, plug them in, and chug out the answers. It’s a reliable shortcut that always works! Examples To solve $x^2 7x + 12 = 0$, use $a=1, b= 7, c=12$: $x = \frac{ ( 7) \pm \sqrt{( 7)^2 4(1)(12)}}{2(1)} = \frac{7 \pm 1}{2}$, so $x=4$ and $x=3$. To solve $2x^2 + 5x 4 = 0$, use $a=2, b=5, c= 4$: $x = \frac{ 5 \pm \sqrt{5^2 4(2)( 4)}}{2(2)} = \frac{ 5 \pm \sqrt{57}}{4}$. To solve $3x^2 + 2x + 5 = 0$, use $a=3, b=2, c=5$: $x = \frac{ 2 \pm \sqrt{2^2 4(3)(5)}}{6} = \frac{ 2 \pm \sqrt{ 56}}{6}$, which has no real solution.
Common Questions
What is the quadratic formula and when do you use it?
The quadratic formula x = (-b ± √(b² - 4ac)) / 2a solves any quadratic equation ax² + bx + c = 0. Use it when factoring is not straightforward or when you need exact decimal or irrational solutions.
What does the discriminant tell you about a quadratic equation?
The discriminant b² - 4ac predicts the number and type of solutions: positive means two real solutions, zero means one repeated real solution, and negative means no real solutions (complex roots). Calculate it before applying the full formula.
How do you avoid errors when applying the quadratic formula?
Write the equation in standard form ax² + bx + c = 0 first, then carefully identify a, b, and c with their signs. Compute b² - 4ac inside the radical before simplifying, and remember the ± produces two separate answers.