To determine which quadratic equations will have two real solutions, we need to calculate the discriminant ($\Delta$) for each equation. The discriminant formula is:

$\Delta = b^2 - 4ac$

• If $\Delta > 0$, the quadratic equation has two real and distinct solutions (or two $x$-intercepts).
• If $\Delta = 0$, the quadratic equation has one real solution (or one $x$-intercept).
• If $\Delta < 0$, the quadratic equation has no real solutions (no $x$-intercepts).

Now, let's calculate the discriminant for each equation.

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1. $0 = 2x^2 - 7x - 9$

Here, $a = 2$, $b = -7$, and $c = -9$.

$\Delta = (-7)^2 - 4(2)(-9) = 49 + 72 = 121$

Since $\Delta = 121 > 0$, this equation has two real solutions.

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2. $0 = x^2 - 4x + 4$

Here, $a = 1$, $b = -4$, and $c = 4$.

$\Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0$

Since $\Delta = 0$, this equation has one real solution.

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3. $0 = 4x^2 - 3x - 1$

Here, $a = 4$, $b = -3$, and $c = -1$.

$\Delta = (-3)^2 - 4(4)(-1) = 9 + 16 = 25$

Since $\Delta = 25 > 0$, this equation has two real solutions.

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4. $0 = x^2 - 2x - 8$

Here, $a = 1$, $b = -2$, and $c = -8$.

$\Delta = (-2)^2 - 4(1)(-8) = 4 + 32 = 36$

Since $\Delta = 36 > 0$, this equation has two real solutions.

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5. $0 = 3x^2 + 5x + 3$

Here, $a = 3$, $b = 5$, and $c = 3$.

$\Delta = (5)^2 - 4(3)(3) = 25 - 36 = -11$

Since $\Delta = -11 < 0$, this equation has no real solutions.

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Summary:

The quadratic equations with two real solutions are:

• $0 = 2x^2 - 7x - 9$
• $0 = 4x^2 - 3x - 1$
• $0 = x^2 - 2x - 8$

Would you like more details or clarifications?

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Here are 5 related questions:

1. What happens if the discriminant is negative in a quadratic equation?
2. How can the quadratic formula be used to find real solutions?
3. How does the graph of a quadratic equation relate to its discriminant?
4. Can a quadratic equation with a discriminant of zero have complex solutions?
5. What is the relationship between the coefficients and the shape of the parabola?

Tip: The discriminant not only tells the number of solutions but also helps determine the nature of the solutions—whether they are real or complex.