Question

solve the equation for all real solutions in simplest form. 4z² + 11z + 3 = -1

Explanation:

Step1: Rewrite to standard quadratic form

Add 1 to both sides.
$4z^2 + 11z + 3 + 1 = 0$
$4z^2 + 11z + 4 = 0$

Step2: Identify coefficients for quadratic formula

For $az^2+bz+c=0$, $a=4$, $b=11$, $c=4$.

Step3: Apply quadratic formula

Use $z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
$$z=\frac{-11\pm\sqrt{11^2-4\times4\times4}}{2\times4}$$

Step4: Calculate discriminant

Compute value under square root.
$\sqrt{121-64}=\sqrt{57}$

Step5: Simplify the expression

Substitute discriminant back.
$z=\frac{-11\pm\sqrt{57}}{8}$

Answer:

$z=\frac{-11+\sqrt{57}}{8}$ and $z=\frac{-11-\sqrt{57}}{8}$