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2022-01-14

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:

Answer & Explanation

Vasquez

Expert2022-02-06Added 669 answers

Discriminant:

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

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

$D = b^{2} - 4 a c$

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

Real Solution:

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

Using the quadratic formula, solve for x.
$x = \frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \times 8}}{2} = \frac{4 \pm \sqrt{16 - 32}}{2} = \frac{4 \pm \sqrt{- 16}}{2} :$
$x = \frac{4 + \sqrt{- 16}}{2} or x = \frac{4 - \sqrt{- 16}}{2}$

Express $\sqrt{- 16}$ in terms of i.
$\sqrt{- 16} = \sqrt{- 1} \sqrt{16} = i \sqrt{16} :$
$x = \frac{4 + i \sqrt{16}}{2} or x = \frac{4 - i \sqrt{16}}{2}$

Simplify radicals.
$\sqrt{16} = \sqrt{2^{4}} = 4 = 4 :$
$x = \frac{4 + i \times 4}{2} or x = \frac{4 - i \times 4}{2}$

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

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

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

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

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