Solve Quadratic Equations by Factoring
This is a tutorial questions on how to solve quadratic equations by factoring. The detailed solutions to thsese questions are included. There is also Factor Quadratic Expressions - Step by Step Calculator in this website.
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Questions with SolutionsQuestion 1
Solve the following quadratic equation.
x 2 - 3x = 0
Solution to Question1
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Given
x 2 - 3x = 0
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Factor x out in the expression on the left.
x (x - 3) = 0
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For the product x (x - 3) to be equal to zero we nedd to have
x = 0 or x - 3 = 0
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Solve the above simple equations to obtain the solutions.
x = 0
or
x = 3
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As an exercise, check that x = 0 and x = 3 are solutions to the given equation.
Question 2
Solve the quadratic equation given below
x 2 - 5 x + 6 = 0
Solution to Question2
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To factor the expression on the left, we need to write x 2 - 5 x + 6 in the form factored:
x 2 - 5 x + 6 = (x + a)(x + b)
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so that the sum of a and b is -5 and their product is 6. The numbers that satisfy these conditions are - 2 and - 3. Hence
x 2 - 5 x + 6 = (x - 2)(x - 3)
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Substitute into the original equation and solve.
(x - 2)(x - 3) = 0
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(x - 2)(x - 3) is equal to zero if
x - 2 = 0
or
x - 3 = 0
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Solve the above equations to obtain two solutions to the given equation.
x = 2
or
x = 3
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As an exercise, check that x = 0 and x = 3 are solutions to the given equation.
Question 3
Solve the following equation
2 x 2 + x - 21 = 0
Solution to Question3
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We first try to write 2 x 2 + x - 21 in the factored form
2 x 2 + x - 21 = (2x + a)(x + b)
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Such that the product a b is equal to - 21 and a + 2 b = 1
two pairs of numbers gives a product of - 21: either -3 and 7 or 3 and -7. After some trial exercises it found that 2 x 2 + x - 21 may be factored as follows:
2 x 2 + x - 21 = (2x + 7)(x - 3)
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We now substitute into the original equation
(2x + 7)(x - 3) = 0
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and solve the following simpler equations
2x + 7 = 0
x - 3 = 0
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to obtain
x = - 7 / 2
or x = 3
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As an exercise, check that x = 0 and x = 3 are solutions to the given equation.
Question 4
Solve the following equation
(x - 1)(x + 1 / 2) = - x + 1
Solution to Question4
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At first we might be tempted into expanding the left side of the equation. However after examination of the right side, the above equation may be written as:
(x - 1)(x + 1 / 2) = - (x - 1)
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Write the equation with the right side equal to zero.
(x - 1)(x + 1 / 2) + (x - 1) = 0
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We now factor (x - 1) out.
(x - 1)(x + 1 / 2 + 1) = 0
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and solve the following simpler equations
x - 1 = 0
x + 3 / 2 = 0
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to obtain
x = 1
or
x = - 3 / 2
More References and linksQuadratic Equations Calculator and Solver.
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