To solve quadratic equations using the quadratic formula, you first need to know the standard form of a quadratic equation, which is:
\[ ax^2 + bx + c = 0 \]
Where:
- \( a \), \( b \), and \( c \) are coefficients (with \( a \neq 0 \)),
- \( x \) represents the variable.
The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here’s a step-by-step guide on how to use the quadratic formula:
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Identify coefficients: Determine the values of \( a \), \( b \), and \( c \) from your quadratic equation.
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Calculate the discriminant: Compute the value of the discriminant, which is the expression under the square root in the formula: \[ D = b^2 - 4ac \]
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Determine the nature of the roots:
- If \( D > 0 \): There are two distinct real roots.
- If \( D = 0 \): There is exactly one real root (a repeated root).
- If \( D < 0 \): There are no real roots (but two complex roots).
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Apply the quadratic formula:
- Substitute \( a \), \( b \), and \( D \) into the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \]
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Calculate the roots:
- If \( D \) is positive, calculate both roots: \[ x_1 = \frac{-b + \sqrt{D}}{2a} \] \[ x_2 = \frac{-b - \sqrt{D}}{2a} \]
- If \( D = 0 \), calculate the single root: \[ x = \frac{-b}{2a} \]
- If \( D < 0 \), express the roots in terms of imaginary numbers: \[ x_1 = \frac{-b + i\sqrt{-D}}{2a} \] \[ x_2 = \frac{-b - i\sqrt{-D}}{2a} \]
Example
Let's solve the quadratic equation:
\[ 2x^2 - 4x - 6 = 0 \]
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Identify coefficients: Here, \( a = 2 \), \( b = -4 \), and \( c = -6 \).
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Calculate the discriminant: \[ D = (-4)^2 - 4 \cdot 2 \cdot (-6) = 16 + 48 = 64 \]
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Determine the nature of the roots: Since \( D > 0 \), we will have two distinct real roots.
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Apply the quadratic formula: \[ x = \frac{-(-4) \pm \sqrt{64}}{2 \cdot 2} = \frac{4 \pm 8}{4} \]
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Calculate the roots:
- First root: \[ x_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3 \]
- Second root: \[ x_2 = \frac{4 - 8}{4} = \frac{-4}{4} = -1 \]
So the solutions to the equation \( 2x^2 - 4x - 6 = 0 \) are \( x = 3 \) and \( x = -1 \).