To solve the quadratic equation \(3A^2 + 5A = 2A^2 - 3A + 2\), we first rearrange the equation to set it to zero.
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Move all terms to one side of the equation: \[ 3A^2 + 5A - 2A^2 + 3A - 2 = 0 \]
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Combine like terms: \[ (3A^2 - 2A^2) + (5A + 3A) - 2 = 0 \] \[ A^2 + 8A - 2 = 0 \]
Now we have the quadratic equation: \[ A^2 + 8A - 2 = 0 \]
We can solve this using the quadratic formula: \[ A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = 8\), and \(c = -2\).
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Calculate the discriminant: \[ b^2 - 4ac = 8^2 - 4 \cdot 1 \cdot (-2) = 64 + 8 = 72 \]
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Substitute in the quadratic formula: \[ A = \frac{-8 \pm \sqrt{72}}{2 \cdot 1} = \frac{-8 \pm 6\sqrt{2}}{2} \]
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Simplify: \[ A = \frac{-8}{2} \pm \frac{6\sqrt{2}}{2} \] \[ A = -4 \pm 3\sqrt{2} \]
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Now, calculate the two possible values for \(A\): \[ A_1 = -4 + 3\sqrt{2} \quad \text{and} \quad A_2 = -4 - 3\sqrt{2} \]
Calculating these to two decimal places:
- Calculate \(3\sqrt{2} \approx 3 \times 1.414 = 4.242\):
- \(A_1 \approx -4 + 4.242 \approx 0.242\)
- \(A_2 \approx -4 - 4.242 \approx -8.242\)
So the solutions to the quadratic equation \(3A^2 + 5A = 2A^2 - 3A + 2\) to two decimal places are: \[ A_1 \approx 0.24 \quad \text{and} \quad A_2 \approx -8.24 \]
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