Question

4. Add the proper constant to the binomial so that the resulting trinomial is a perfect square trinomial. Then factor the

Answer 1

Answer:

The constant is 289/4

The factored form is (x + 17/2)²

Step-by-step explanation:

Perfect square trinomials are in standard form ax² + bx + c; when the square root of "ax²", multiplied by the square root of "c", multiplied by 2, is equal to "bx". In equation form:

2√(ax²)√c = bx

In x² + 17x + c:

ax² = x²

bx = 17x

c = the constant term we are finding

Substitute what we know into the equation, then find c.

2√(ax²)√c = bx

2√(x²)√c = 17x        Simplify √(x²) = x because √ and ² cancel out

2x√c = 17x

√c = 17x / 2x         Divide both sides by 2x

√c = 17/2            Keep as a fraction to get the exact value of "c"

c = (17/2)²         Square both sides and simplify

c = 289/4

Therefore the constant term is 289/4.

To factor a perfect trinomial, use the form:

ax² + bx + c = ( √(ax²) ± √c )²

Take the square root of the first term, plus/minus the square root of the last term, then square the entire equation.

Whether the middle sign is plus (+) or minus (-) depends on if bx is positive or negative.

The perfect trinomial is x² + 17x +  289/4 after having substituted "c".

Square root the first term, plus square root the last term and all squared:

( √(x²) + √(289/4) )²   Simplify. Find the square root of the numbers.

= (x + 17/2)²

The factored form is (x + 17/2)².