```Question 83944
Given:
.
{{{5x^2 - x^2 = 42}}}
.
If you want, you can solve this without using the quadratic formula.  Notice that the left
side of this equation simplifies to:
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{{{5x^2 - x^2 = 4x^2}}}
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So you can substitute {{{4x^2 }}} for the left side of the equation you were given. This
makes the given equation become:
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{{{4x^2 = 42}}}
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Then you can find {{{x^2}}} by dividing both sides by 4 which is the multiplier of the
{{{x^2}}} term.  This division of both sides by 4 reduces the equation to:
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{{{x^2 = 42/4}}}
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and {{{42/4 = 10.5}}}
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Substituting {{{10.5}}} for {{{42/4}}} results in the equation becoming:
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{{{x^2 = 10.5}}}
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and taking the square root of both sides gives you the two answers:
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{{{x = sqrt(10.5)}}} and {{{x = -sqrt(10.5)}}}
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And since the square root of 10.5 is (from a calculator) 3.240370349, the two answers to
this problem are:
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{{{x = 3.240370349}}} and {{{x = -3.240370349}}}
.
But if you have to do the problem using the quadratic formula, you need to get the
original problem into the standard quadratic form of:
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{{{ax^2 + bx + c = 0}}}
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You still combine the two terms on the left side of the original equation to get:
.
{{{4x^2 = 42}}}
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just as we did above.  Next you need to get zero on the right side as the standard
quadratic form calls for. Do this by subtracting 42 from both sides of the equation.
That subtraction results in:
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{{{4x^2 - 42 = 0}}}
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This is in the standard quadratic form in which a = 4, b = 0 (because there is no term
that contains just x), and c = -42.
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The quadratic formula says that the answer to the problem is determined by the equation:
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{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
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Since b = 0 in this problem, the equation reduces to:
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{{{x = (0 +- sqrt( 0-4*a*c ))/(2*a) }}}
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Next substitute 4 for a and -42 for c to get:
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{{{x = (0 +- sqrt( 0-4*4*(-42) ))/(2*4) }}}
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The term in the radical [that is -4*4*(-42)] multiplies out to be 672 so the problem becomes:
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{{{x = (0 +- sqrt( 672 ))/(2*4) }}}
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The denominator of (2*4) becomes 8 which further simplifies the problem to:
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{{{x = (0 +- sqrt(672))/8}}}
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Since the square root of 672 is (by a calculator) 25.92296279 the answers to this
problem become:
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{{{x = 25.92296279/8}}} and {{{x = -25.92296279/8}}}
.
When you do the division, the two answers are:
.
{{{x = 3.240370349}}} and {{{x = -3.240370349}}}
.
And these are the same answers that we found earlier without using the quadratic formula.
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Hope this discussion helps you to work your way through the problem.```