Lesson 9.3: Solve Quadratic Equations Using the Quadratic Formula

New Concept

The Quadratic Formula is a universal tool for solving any quadratic equation of the form $ax^2 + bx + c = 0$. Derived from completing the square, it provides a direct path to the solutions by using the coefficients $a, b,$ and $c$.

What’s next

You'll now see how to apply the formula through worked examples and interactive practice cards. Soon, you'll use it to tackle challenge problems.

Property

The solutions to a quadratic equation of the form $ax^2 + bx + c = 0$, where $a \neq 0$ are given by the formula:

To solve a quadratic equation using the Quadratic Formula:
Step 1. Write the quadratic equation in standard form, $ax^2 + bx + c = 0$. Identify the values of $a$, $b$, and $c$.
Step 2. Write the Quadratic Formula. Then substitute in the values of $a$, $b$, and $c$.
Step 3. Simplify.
Step 4. Check the solutions.

Examples

• To solve $2x^2 + 5x - 3 = 0$, we identify $a=2, b=5, c=-3$. Substituting into the formula gives $x = \frac{-5 \pm \sqrt{5^2 - 4(2)(-3)}}{2(2)} = \frac{-5 \pm \sqrt{25 + 24}}{4} = \frac{-5 \pm \sqrt{49}}{4} = \frac{-5 \pm 7}{4}$. The solutions are $x = \frac{1}{2}$ and $x = -3$.
• To solve $3x^2 + 10x + 5 = 0$, we have $a=3, b=10, c=5$. The formula gives $x = \frac{-10 \pm \sqrt{10^2 - 4(3)(5)}}{2(3)} = \frac{-10 \pm \sqrt{100 - 60}}{6} = \frac{-10 \pm \sqrt{40}}{6} = \frac{-10 \pm 2\sqrt{10}}{6} = \frac{-5 \pm \sqrt{10}}{3}$.
• To solve $x^2 + 2x + 10 = 0$, we have $a=1, b=2, c=10$. The formula gives $x = \frac{-2 \pm \sqrt{2^2 - 4(1)(10)}}{2(1)} = \frac{-2 \pm \sqrt{4 - 40}}{2} = \frac{-2 \pm \sqrt{-36}}{2} = \frac{-2 \pm 6i}{2} = -1 \pm 3i$.

Explanation

The Quadratic Formula is a powerful tool derived from completing the square on the general quadratic equation. It provides a direct solution for any quadratic equation, saving you from repeating the steps of completing the square every time.

Property

In the Quadratic Formula, $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, the quantity $b^2 - 4ac$ is called the discriminant.
For a quadratic equation of the form $ax^2 + bx + c = 0$, $a \neq 0$:

• If $b^2 - 4ac > 0$, the equation has 2 real solutions.
• If $b^2 - 4ac = 0$, the equation has 1 real solution.
• If $b^2 - 4ac < 0$, the equation has 2 complex solutions.

Examples

• For the equation $3x^2 + 5x - 8 = 0$, the discriminant is $(5)^2 - 4(3)(-8) = 25 + 96 = 121$. Since $121 > 0$, there are 2 real solutions.
• For the equation $4y^2 - 28y + 49 = 0$, the discriminant is $(-28)^2 - 4(4)(49) = 784 - 784 = 0$. Since the discriminant is 0, there is 1 real solution.
• For the equation $2n^2 + 3n + 5 = 0$, the discriminant is $(3)^2 - 4(2)(5) = 9 - 40 = -31$. Since $-31 < 0$, there are 2 complex solutions.

Explanation

The discriminant is the part of the Quadratic Formula under the radical. Its value predicts the number and type of solutions without having to solve the entire equation. A positive, zero, or negative result tells you exactly what kind of answers to expect.

Property

To identify the most appropriate method to solve a quadratic equation:
Step 1. Try Factoring first. If the quadratic factors easily, this method is very quick.
Step 2. Try the Square Root Property next. If the equation fits the form $ax^2 = k$ or $a(x - h)^2 = k$, it can easily be solved by using the Square Root Property.
Step 3. Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

Examples

• For $x^2 - 8x + 15 = 0$, we can easily factor it as $(x-3)(x-5)=0$. The most appropriate method is Factoring.
• For $2(x-5)^2 = 32$, the equation is in the form $a(x-h)^2=k$. The most appropriate method is the Square Root Property.
• For $7u^2 + 5u - 1 = 0$, trial and error factoring is difficult. The most appropriate method is the Quadratic Formula.

Explanation

This strategy helps you choose the most efficient tool for solving a quadratic equation. Start with the fastest method, factoring. If that is not practical, check if the Square Root Property applies. The Quadratic Formula is the universal method that works for all cases.