# &nbsp; Find the discriminant&nbsp;b2&minus;4acb2-4ac, and use it to determine

Find the discriminant b2−4acb2-4ac, and use it to determine the number of real solutions to the equation. a2+8=4aa2+8=4a Discriminant: Real Solution:
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Discriminant:

${a}^{2}+8=4a$

${a}^{2}-4a+8=0$

$D={b}^{2}-4ac$

$=\left(-4{\right)}^{2}-4\cdot 1\cdot 8=16-32=-16$ - Answer

Real Solution:

Solve for $x$:
${x}^{2}-4x+8=0$

Using the quadratic formula, solve for x.
$x=\frac{4±\sqrt{{\left(-4\right)}^{2}-4×8}}{2}=\frac{4±\sqrt{16-32}}{2}=\frac{4±\sqrt{-16}}{2}:$
$x=\frac{4+\sqrt{-16}}{2}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x=\frac{4-\sqrt{-16}}{2}$

Express $\sqrt{-16}$ in terms of i.
$\sqrt{-16}=\sqrt{-1}\sqrt{16}=i\sqrt{16}:$
$x=\frac{4+i\sqrt{16}}{2}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x=\frac{4-i\sqrt{16}}{2}$

$\sqrt{16}=\sqrt{{2}^{4}}=4=4:$
$x=\frac{4+i×4}{2}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x=\frac{4-i×4}{2}$

Factor the greatest common divisor (gcd) of $4,4i$ and 2 from $4+4i$.
Factor 2 from $4+4i$ giving $4+4i:$
$x=\frac{1}{2}4+4$ i or $x=\frac{4-4i}{2}$

Cancel common terms in the numerator and denominator.
$\frac{4+4i}{2}=2+2i:$
$x=2+2i$ or $x=\frac{4-4i}{2}$

Factor the greatest common divisor (gcd) of $4,-4i$ and 2 from $4-4i.$
Factor 2 from $4-4i$ giving $4-4i:$
$x=2+2i$ or $x=\frac{1}{2}4-4i$

Cancel common terms in the numerator and denominator.
$\frac{4-4i}{2}=2-2i:$
$x=2+2i$ or $x=2-2i$ -  Answer