Determine the Nature of the Roots Using the Discriminant 3x^2=8x+7

$3 x^{2} = 8 x + 7$

Move all the expressions to the left side of the equation.

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Move $8 x$ to the left side of the equation by subtracting it from both sides.

$3 x^{2} - 8 x = 7$

Move $7$ to the left side of the equation by subtracting it from both sides.

$3 x^{2} - 8 x - 7 = 0$

The discriminant of a quadratic is the expression inside the radical of the quadratic formula.

$b^{2} - 4 (a c)$

Substitute in the values of $a$ , $b$ , and $c$ .

${(- 8)}^{2} - 4 (3 \cdot - 7)$

Evaluate the result to find the discriminant.

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Simplify each term.

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Raise $- 8$ to the power of $2$ .

$64 - 4 (3 \cdot - 7)$

Multiply $3$ by $- 7$ .

$64 - 4 \cdot - 21$

Multiply $- 4$ by $- 21$ .

$64 + 84$

Add $64$ and $84$ .

$148$

The nature of the roots of the quadratic can fall into one of three categories depending on the value of the discriminant $(Δ)$ :

$Δ > 0$ means there are $2$ distinct real roots.

$Δ = 0$ means there are $2$ equal real roots, or $1$ distinct real root.

$Δ < 0$ means there are no real roots, but $2$ complex roots.

Since the discriminant is greater than $0$ , there are two real roots.

Two Real Roots

Determine the Nature of the Roots Using the Discriminant 3x^2=8x+7

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