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Use the discriminant to determine whether the quadratic equation has two unequal real solutions, a repeated real solution, or no real solution, without solving the equation. $3 x^{2}+7 x+5=0$
Which of the following correctly describes the solutions to the given equation? A. A repeated real solution B. No real solution C. Two unequal real solutions

Studdy Solution

STEP 1

1. The quadratic equation is given in the standard form $ax^2 + bx + c = 0$ .
2. The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by $\Delta = b^2 - 4ac$ .
3. The nature of the roots of the quadratic equation depends on the value of the discriminant: - If $\Delta > 0$ , the equation has two unequal real solutions. - If $\Delta = 0$ , the equation has a repeated real solution. - If $\Delta < 0$ , the equation has no real solution.

STEP 2

1. Identify the coefficients $a$ , $b$ , and $c$ .
2. Calculate the discriminant $\Delta$ .
3. Analyze the discriminant to determine the nature of the solutions.

STEP 3

Identify the coefficients from the quadratic equation $3x^2 + 7x + 5 = 0$ :
- $a = 3$ - $b = 7$ - $c = 5$

STEP 4

Calculate the discriminant $\Delta$ using the formula $\Delta = b^2 - 4ac$ :
$\Delta = 7^2 - 4 \times 3 \times 5$
$\Delta = 49 - 60$
$\Delta = -11$

STEP 5

Analyze the discriminant:
Since $\Delta = -11$ , which is less than 0, the quadratic equation has no real solution.
The correct description of the solutions to the given equation is $\boxed{B. \text{No real solution}}$ .

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