Lesson 35: Solving Quadratic Equations I

New Concept

The zeros of a function $f(x)$ are the $x$-values that make $f(x) = 0$.

What’s next

Next, you'll use the Zero Product Property as your primary tool to find the zeros of quadratic functions by factoring.

The zeros of a function $f(x)$ are the $x$-values that make $f(x) = 0$. To find the zeros of a quadratic function, solve the related equation $ax^2 + bx + c = 0$. The solutions to the equation are also called its roots.

Find the zeros of $f(x) = 5x^2 + x$. First, set to zero: $5x^2 + x = 0$. Next, factor the expression: $x(5x + 1) = 0$. The zeros are $x=0$ or $x = -\frac{1}{5}$.
Find the roots of $x^2 = -3x + 18$. Rearrange into standard form: $x^2 + 3x - 18 = 0$. Factor the quadratic: $(x+6)(x-3) = 0$. The roots are $x=-6$ and $x=3$.

Think of a function's 'zeros' as its secret identity reveal! They are the special x-values where the function's output becomes zero. For quadratics, this means finding where its U-shaped graph crosses the x-axis. Finding these 'roots' is like solving a puzzle to discover which x-values make the entire equation equal zero. It's where the magic happens!

Property

Let $a$ and $b$ be real numbers. If $ab = 0$, then $a = 0$ or $b = 0$.

Examples

When a quadratic function has exactly one real zero, that zero is a double root of the related equation.

To solve $x^2 - 12x + 36 = 0$, we factor it into $(x-6)(x-6)=0$. Since both factors give the same solution, $x=6$ is a double root.
The equation $(x+5)^2 = 0$ means $(x+5)(x+5)=0$. This equation has a double root at $x=-5$, and the graph of $f(x)=(x+5)^2$ touches the x-axis only at that point.

A double root is like a parabola that's too shy to cross the x-axis. Instead, it just gently touches it at one single point and bounces right back! This special event happens when both factors of the quadratic are identical, giving you the same solution twice. It's a unique meeting point where the graph kisses the axis.

To write a quadratic function from its zeros, reverse the solving process. If the zeros are $r_1$ and $r_2$, start with the factored equation $(x - r_1)(x - r_2) = 0$ and expand it into the standard form $ax^2 + bx + c = 0$.

Write a function with zeros $4$ and $-6$. Start with the factors: $(x-4)(x-(-6))=0$, which is $(x-4)(x+6)=0$. Expand it: $x^2 + 6x - 4x - 24 = 0$. The function is $f(x) = x^2 + 2x - 24$.
Write a function with zeros $-\frac{3}{2}$ and $1$. Start with $(x+\frac{3}{2})(x-1)=0$. Multiply by 2 to clear the fraction: $(2x+3)(x-1)=0$. Expand: $2x^2 - 2x + 3x - 3 = 0$. The function is $f(x)=2x^2+x-3$.

This is like being a math magician who builds a quadratic from scratch! If you know the 'ending'—the zeros where the graph hits the x-axis—you can work backward. Just turn those zeros back into factors, multiply them together, and poof! You have perfectly reconstructed the original quadratic function. It’s a reverse puzzle that is super satisfying to solve.