Question
Question Which expression is equivalent to \( 5 x^{2}-18 x+9 \) ?
Ask by Wade Summers. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
An equivalent expression is \( (5x - 3)(x - 3) \).
Solution
To factor the quadratic expression 5x² – 18x + 9, we can proceed by finding its roots.
Step 1. Compute the discriminant:
D = (–18)² – 4(5)(9) = 324 – 180 = 144
Step 2. Since D is a perfect square (144 = 12²), use the quadratic formula to find the roots:
x = [18 ± 12] / (2·5)
x = (18 + 12) / 10 = 30/10 = 3
x = (18 – 12) / 10 = 6/10 = 3/5
Step 3. Write the quadratic in factored form using the roots:
5x² – 18x + 9 = 5(x – 3)(x – 3/5)
To remove the fraction, notice that:
5(x – 3)(x – 3/5) = (5x – 3)(x – 3)
Thus, an expression equivalent to 5x² – 18x + 9 is:
(5x – 3)(x – 3)
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Simplify this solution The Deep Dive
To factor the quadratic expression \( 5x^{2} - 18x + 9 \), you can use the method of grouping or apply the quadratic formula. The expression can be rewritten as \( 5x^{2} - 15x - 3x + 9 \) and then grouped into pairs. This process ultimately leads to factoring the expression as \( (5x + 3)(x - 3) \), which is its equivalent form! Alternatively, you could use the quadratic formula, where you'll replace \( a, b, \) and \( c \) with 5, -18, and 9 respectively. This yields roots that can help you express the polynomial in its factored form!
