Lesson 10.3: Solve Quadratic Equations Using the Quadratic Formula

New Concept

The Quadratic Formula is a powerful shortcut derived from completing the square. It universally solves any quadratic equation, $ax^2 + bx + c = 0$, and its discriminant, $b^2 - 4ac$, predicts how many solutions you'll find.

What’s next

Next, you'll master this formula through a series of interactive examples, practice cards, and challenge problems to solidify your skills.

Property

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

To solve an equation using this formula, first write it in standard form to identify the values of $a, b,$ and $c$.
Next, substitute these values into the formula.
Finally, simplify the expression to find the solution(s).

Examples

• To solve $x^2 - 3x - 10 = 0$, use $a=1, b=-3, c=-10$. Then $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-10)}}{2(1)} = \frac{3 \pm \sqrt{49}}{2} = \frac{3 \pm 7}{2}$. The solutions are $x=5$ and $x=-2$.
• To solve $3x^2 + 5x - 1 = 0$, use $a=3, b=5, c=-1$. Then $x = \frac{-5 \pm \sqrt{5^2 - 4(3)(-1)}}{2(3)} = \frac{-5 \pm \sqrt{25+12}}{6} = \frac{-5 \pm \sqrt{37}}{6}$.
• To solve $x(x-4) = -2$, first rewrite it as $x^2 - 4x + 2 = 0$. With $a=1, b=-4, c=2$, we get $x = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(2)}}{2} = \frac{4 \pm \sqrt{8}}{2} = 2 \pm \sqrt{2}$.

Explanation

The Quadratic Formula is your ultimate tool for solving any quadratic equation. It is derived from completing the square, giving you a direct plug-and-play method to find solutions just by using the coefficients $a, b$, and $c$.

Property

In the Quadratic Formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, 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 solutions.
• if $b^2 - 4ac = 0$, the equation has 1 solution.
• if $b^2 - 4ac < 0$, the equation has no real solutions.

Examples

• For $x^2 - 6x + 7 = 0$, the discriminant is $(-6)^2 - 4(1)(7) = 36 - 28 = 8$. Since $8 > 0$, there are two real solutions.
• For $9x^2 + 6x + 1 = 0$, the discriminant is $6^2 - 4(9)(1) = 36 - 36 = 0$. Since it is zero, there is exactly one real solution.
• For $4x^2 - 3x + 2 = 0$, the discriminant is $(-3)^2 - 4(4)(2) = 9 - 32 = -23$. Since $-23 < 0$, there are no real solutions.

Explanation

The discriminant is the part inside the square root of the Quadratic Formula. Its value tells you the number of real solutions without having to solve the whole equation. It's a quick way to predict the outcome!

Property

1. Step 1. Try Factoring first. If the quadratic factors easily, this method is very quick.
2. 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.
3. Step 3. Use the Quadratic Formula. Any quadratic equation can be solved by using the Quadratic Formula.

Examples

• For $x^2 - 9x + 20 = 0$, Factoring is best. The equation factors into $(x-4)(x-5)=0$, giving solutions $x=4$ and $x=5$ quickly.
• For $2(x-3)^2 = 32$, the Square Root Property is ideal. Divide by 2 to get $(x-3)^2=16$, then take the square root: $x-3 = \pm 4$, so $x=7$ or $x=-1$.
• For $5x^2 + 7x - 11 = 0$, the numbers do not allow for easy factoring. The Quadratic Formula is the most appropriate and reliable method to find the solutions.

Explanation

While the Quadratic Formula is a universal solver, it is not always the fastest. Always check if an equation can be factored or if the Square Root Property applies. Choosing the right tool makes solving equations much simpler.