Solve 4x² + 16x + 7 = 0.

Step-by-step solution

1. 1

   Compute the discriminant first. With $a=4$, $b=16$, $c=7$,

   $b^{2}-4ac=16^{2}-4(4)(7)=256-112=144$

   A positive perfect square means two distinct rational roots — so both the quadratic formula and factoring will work, and the formula is the more mechanical route.

2. 2

   Apply the quadratic formula.

   $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{-16\pm\sqrt{144}}{8}=\frac{-16\pm 12}{8}$

3. 3

   Evaluate the two branches.

   $x=\frac{-16+12}{8}=\frac{-4}{8}=-\frac12,\qquad x=\frac{-16-12}{8}=\frac{-28}{8}=-\frac72$

   $\boxed{x=-\tfrac12\ \text{or}\ x=-\tfrac72}$

4. 4

   Cross-check by factoring. Because the discriminant was a perfect square, the quadratic factors over the rationals:

   $4x^{2}+16x+7=(2x+1)(2x+7)$

   Expanding: $4x^{2}+14x+2x+7=4x^{2}+16x+7$ ✓, and setting each bracket to zero gives the same two roots.

5. 5

   Check with Vieta's formulas. The roots must sum to $-b/a=-4$ and multiply to $c/a=\tfrac74$: $-\tfrac12-\tfrac72=-4$ ✓ and $\left(-\tfrac12\right)\left(-\tfrac72\right)=\tfrac74$ ✓. Both signs being negative is also expected, since $b$ and $c$ are both positive with $a>0$.