Question
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Tutor-Verified Answer
Answer
The solutions are \( x = 3 \) and \( x = 5 \).
Solution
**Step 1.** Write down the given quadratic equation:
\[
x^2 - 8x + 15 = 0
\]
**Step 2.** Factor the quadratic expression. We need two numbers that multiply to \(15\) (the constant term) and add up to \(-8\) (the coefficient of \(x\)). The numbers \(-3\) and \(-5\) satisfy these conditions since:
\[
-3 \times -5 = 15 \quad \text{and} \quad -3 + (-5) = -8
\]
**Step 3.** Factor the quadratic:
\[
x^2 - 8x + 15 = (x - 3)(x - 5)
\]
**Step 4.** Set each factor equal to zero and solve for \(x\):
\[
x - 3 = 0 \quad \text{or} \quad x - 5 = 0
\]
**Step 5.** Solve the equations:
\[
\text{If } x - 3 = 0, \quad \text{then } x = 3.
\]
\[
\text{If } x - 5 = 0, \quad \text{then } x = 5.
\]
**Final Answer:**
\[
x = 3 \quad \text{or} \quad x = 5.
\]
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Simplify this solution The Deep Dive
This quadratic equation can be solved by factoring. It can be rewritten as \( (x-3)(x-5)=0 \). From this, we find the solutions \( x = 3 \) and \( x = 5 \). These values are the points where the graph of the equation intersects the x-axis! Quadratic equations like this one have real-world applications, especially in areas such as physics, engineering, and finance. For instance, they can model projectile motion, where the height of an object over time follows a parabola. So, whether you’re launching a rocket or calculating profit margins, you’re likely to encounter these quadratic beauties!
