1Cademy - Finding Points on the Graph of $$f(x) = x^2 - 2x

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Zero Product Property

Example

Finding Points on the Graph of $f(x) = x^2 - 2x - 8$ when $f(x) = 7$

To find $x$ when $f(x) = 7$ for the polynomial function $f(x) = x^2 - 2x - 8$ , substitute $7$ for $f(x)$ to set up the equation $7 = x^2 - 2x - 8$ . Rewrite the quadratic equation in standard form by subtracting $7$ from both sides to get $x^2 - 2x - 15 = 0$ . Factor the trinomial into $(x - 5)(x + 3) = 0$ . Use the Zero Product Property to set each factor to zero: $x - 5 = 0$ or $x + 3 = 0$ , which yields $x = 5$ and $x = -3$ . Thus, the two points $(5, 7)$ and $(-3, 7)$ lie on the graph of the function.

Updated 2026-04-30

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OpenStax Intermediate Algebra

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