This calculator solves quadratic equations by completing the square or by using quadratic formula. It displays the work process and the detailed explanation. Every step will be explained in detail.
A general quadratic equation can be written in the form $ax^2 + bx + c = 0$. This calculator solves quadratic equation using two methods.
When $a \ne 0$ , there are two solutions to $ax^2 + bx + c = 0$ and they are
$$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}.$$The formula can be used to solve any quadratic equation.
Example:
Solve equation $2x^2 + 7x - 15 = 0$ using the quadratic formula.
Solution:
Here we have : $ a = 2 ~ b = 7 ~ c = -15 $
To calculate first solution we use "+" sign: |
To calculate second solution we use "-" sign: |
$$x_{1} = \frac{-7 + 13} {4}$$
$$x_{1} = \frac{6}{4}$$
$$x_{1} = \frac{3}{2}$$
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$$x_{2} = \frac{-7 - 13} {4}$$
$$x_{2} = \frac{-20} {4}$$
$$x_{2} = -5$$
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Exercise:
Solve equation 3x2 + 2x - 5 = 0. ( Use above calculator to check your solution. )
The best way to learn this method is by using an example.
Example:
Solve equation 2x2 + 7x - 15 = 0 by completing the square.
Solution:
$$2x^2 + 7x - 15 = 0$$
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$$2x^2 + 7x - 15 = 0 / : 2$$
$$x^2 + \frac{7}{2}x - \frac{15}{2} = 0$$
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Step1: Divide all terms by the coefficient of x2. |
$$x^2 + \frac{7}{2}x = \frac{15}{2}$$
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Step 2: Keep all terms containing x on one side. Move the constant to the right. |
$$x^2 + \frac{7}{2}x + {\left(\frac{7}{4}\right)}^2= \frac{15}{2}+{\left(\frac{7}{4}\right)}^2$$
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Step 3: Take half of the x-term coefficient and square it. Add this value to both sides. |
$$x^2 + \frac{7}{2}x + {\left(\frac{7}{4}\right)}^2= \frac{169}{16}$$
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Step 4: Simplify right side. |
$$ {\left(x + \frac{7}{4}\right)}^2= \frac{169}{16}$$
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Step 5: Write the perfect square on the left. |
$$ x + \frac{7}{4}= \pm\sqrt{\frac{169}{16}}$$
$$ x + \frac{7}{4}= \pm\frac{13}{4}$$
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Step 6: Take the square root on both sides of the equation. |
$$ x_1 = - \frac{7}{4} + \frac{13}{4} = \frac{3}{2}$$
$$ x_2 = - \frac{7}{4} - \frac{13}{4} = -5$$
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Step 7: Solve for x. |
Exercise:
Solve equation x2 - 4x + 3 = 0. ( Use above calculator to check your solution. )
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