Question

select the factored form of each expression.
16x²-8x+1 =
x²-x- 30 =
9x² - 49 =
3x² + 17x - 6 =
done

Explanation:

Step1: Factor \(16x^2 - 8x + 1\)

This is a perfect square trinomial. The formula for \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 4x\) (since \((4x)^2 = 16x^2\)) and \(b = 1\) (since \(1^2 = 1\)), and \(2ab = 2\times4x\times1 = 8x\), which matches the middle term. So \(16x^2 - 8x + 1=(4x - 1)^2\).

Step2: Factor \(x^2 - x - 30\)

We need two numbers that multiply to \(-30\) and add to \(-1\). The numbers are \(-6\) and \(5\) (since \(-6\times5=-30\) and \(-6 + 5=-1\)). So \(x^2 - x - 30=(x - 6)(x + 5)\).

Step3: Factor \(9x^2 - 49\)

This is a difference of squares. The formula for \(a^2 - b^2=(a - b)(a + b)\). Here, \(a = 3x\) (since \((3x)^2 = 9x^2\)) and \(b = 7\) (since \(7^2 = 49\)). So \(9x^2 - 49=(3x - 7)(3x + 7)\).

Step4: Factor \(3x^2 + 17x - 6\)

We need two numbers that multiply to \(3\times(-6)=-18\) and add to \(17\). The numbers are \(18\) and \(-1\). Rewrite the middle term: \(3x^2+18x - x - 6\). Group the terms: \((3x^2 + 18x)-(x + 6)=3x(x + 6)-1(x + 6)=(3x - 1)(x + 6)\).

Answer:

• \(16x^2 - 8x + 1=(4x - 1)^2\)
• \(x^2 - x - 30=(x - 6)(x + 5)\)
• \(9x^2 - 49=(3x - 7)(3x + 7)\)
• \(3x^2 + 17x - 6=(3x - 1)(x + 6)\)