Quadratic Formula

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What does the discriminant indicate in a quadratic equation?

Back

The discriminant indicates the nature of the roots of a quadratic equation. If it is greater than 0, there are two distinct real solutions; if it is equal to 0, there is one real solution; and if it is less than 0, there are no real solutions.

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What is the quadratic formula?

Back

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), used to find the solutions of a quadratic equation ax² + bx + c = 0.

Front

How do you calculate the discriminant?

Back

The discriminant is calculated using the formula b² - 4ac, where a, b, and c are the coefficients of the quadratic equation.

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What does it mean if the discriminant is 0?

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If the discriminant is 0, it means that the quadratic equation has exactly one real solution (a repeated root).

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How can you find the roots of the equation 2x² + 2x - 12 using the quadratic formula?

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To find the roots, substitute a=2, b=2, and c=-12 into the quadratic formula: x = (-2 ± √(2² - 4(2)(-12))) / (2(2)). This simplifies to x = 2 and x = -3.

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What are the steps to solve a quadratic equation using the quadratic formula?

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1. Identify coefficients a, b, and c from the equation. 2. Calculate the discriminant (b² - 4ac). 3. Substitute a, b, and the discriminant into the quadratic formula. 4. Simplify to find the values of x.

Front

What is the significance of the coefficients a, b, and c in a quadratic equation?

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In a quadratic equation ax² + bx + c = 0, 'a' determines the direction and width of the parabola, 'b' affects the position of the vertex, and 'c' represents the y-intercept.

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How do you determine the number of real solutions from the discriminant?

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By evaluating the discriminant: if it's positive, there are two real solutions; if it's zero, there is one real solution; if it's negative, there are no real solutions.

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What is the vertex form of a quadratic equation?

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The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

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How can you convert a quadratic equation from standard form to vertex form?

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To convert from standard form (ax² + bx + c) to vertex form (a(x - h)² + k), complete the square.