Question

Factor the expression. Tell which special product factoring pattern you used. $$4 b^{2}-40 b+100$$

Short answer

The factored form of the expression \(4b^{2} - 40b + 100\) is \((2b - 10)^{2}\). It was identified as a perfect square trinomial.

Step-by-step solution

Identify the quadratic form

Recognize that the equation is in the quadratic form, which is \(ax^{2} + bx + c\). We have \(a = 4\), \(b = -40\), and \(c = 100\). Its square roots will be \(\sqrt{a}\) and \(\sqrt{c}\).

Check if it's a perfect square trinomial

Check whether the given trinomial is perfect square or not. For a trinomial \(ax^{2} + bx + c\) to be a perfect square trinomial, \(2\cdot\sqrt{a}\cdot\sqrt{c}\) should be equal to \(b\). So, if \(2\cdot\sqrt{4}\cdot\sqrt{100} = -40\), the equation is a perfect square trinomial.

Factor the trinomial

If conditions in Step 2 are satisfied, it implies that the given trinomial is a perfect square trinomial and it can be written as \((\sqrt{a}x-\sqrt{c})^{2}\). Hence, the given expression \(4b^{2} - 40b + 100\) is equal to \((2b - 10)^{2}\)