-(1)/(2)(t-3)+t^(2)=0 How many distinct real solutions does the given equation have? Choose 1 answer: (A) 0 (B) 1 (c) 2 (D) 4

Write equation: Write down the given equation. The equation is: -(1)/(2)(t-3) + t^2 = 0Multiply by -2: Multiply both sides of the equation by -2 to get rid of the fraction. -2 * [-(1)/(2)(t-3) + t^2] = -2 * 0 This simplifies to: (t-3) - 2t^2 = 0Rearrange to quadratic form: Rearrange the equation to standard quadratic form. -2t^2 + t - 3 = 0Use discriminant: Use the discriminant to determine the number of real solutions. The discriminant of a quadratic equation ax^2 + bx + c is b^2 - 4ac. For the equation -2t^2 + t - 3, a = -2, b = 1, and c = -3. Discriminant = b^2 - 4ac = (1)^2 - 4(-2)(-3) = 1 - 24 = -23Interpret discriminant: Interpret the discriminant. Since the discriminant is negative (-23), there are no real solutions to the equation.

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-(1)/(2)(t-3)+t^(2)=0
How many distinct real solutions does the given equation have?
Choose 1 answer:
(A) 0
(B) 1
(c) 2
(D) 4

$-\frac{1}{2}(t-3)+t^{2}=0$ $\newline$ How many distinct real solutions does the given equation have? $\newline$ Choose $1$ answer: $\newline$ (A) $0$ $\newline$ (B) $1$ $\newline$ (C) $2$ $\newline$ (D) $4$

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Full solution

-\frac{1}{2}(t-3)+t^{2}=0

\newline

How many distinct real solutions does the given equation have?

\newline

Choose

1

answer:

\newline

(A)

0

\newline

(B)

1

\newline

(C)

2

\newline

(D)

4

Write equation: Write down the given equation. $\newline$ The equation is: $-\frac{1}{2}(t-3) + t^2 = 0$
Multiply by $-2$ : Multiply both sides of the equation by $-2$ to get rid of the fraction. $\newline$ $-2 \cdot \left[-\frac{1}{2}(t-3) + t^2\right] = -2 \cdot 0$ $\newline$ This simplifies to: $(t-3) - 2t^2 = 0$
Rearrange to quadratic form: Rearrange the equation to standard quadratic form. $-2t^2 + t - 3 = 0$
Use discriminant: Use the discriminant to determine the number of real solutions. $\newline$ The discriminant of a quadratic equation $ax^2 + bx + c$ is $b^2 - 4ac$ . $\newline$ For the equation $-2t^2 + t - 3$ , $a = -2$ , $b = 1$ , and $c = -3$ . $\newline$ Discriminant = $b^2 - 4ac = (1)^2 - 4(-2)(-3) = 1 - 24 = -23$
Interpret discriminant: Interpret the discriminant. $\newline$ Since the discriminant is negative $-23$ , there are no real solutions to the equation.