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Prove that two π₯ squared plus two π₯ plus one is always positive.
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To prove a quadratic expression always gives a positive answer, we always complete the square.
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The reason for this will become apparent in a moment.
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When we complete the square for a quadratic expression which has one lot of π₯ squared, it looks like this: π₯ plus π all squared plus π.
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The coefficient for our π₯ squared β thatβs the number in front of the π₯ squared β is two.
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So our completed square form is going to look a little bit different.
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To complete the square for a more complicated expression like this, we need to begin by forming an expression which has a coefficient of π₯ squared of one.
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To do that, we factorise the whole expression by taking out a factor of two.
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Two π₯ squared divided by two is π₯ squared.
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Two π₯ divided by two is π₯.
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And one divided by two is 0.5 or one-half.
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Letβs complete the square for the expression inside our brackets.
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To find the value of the constant inside this first bracket, we halve the coefficient of π₯.
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The coefficient of π₯ is one.
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And a half of one is one-half.
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We then square this number.
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One-half squared is a half multiplied by a half, which is a quarter or 0.25.
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We need our expression to be plus 0.5.
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So to achieve this, weβre going to have to add an additional 0.25 into our completed square form.
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We can say then that, in completed square form, our expression is two lots of π₯ plus 0.5 all squared plus 0.25.
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We can expand this back out, but it isnβt entirely necessary.
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Two lots of π₯ plus 0.5 all squared is just two lots of π₯ plus 0.5 all squared.
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And two lots of 0.25 is 0.5.
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Remember, no matter the value, whenever we square a number, it will always be positive.
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And this is because a positive multiplied by a positive is positive, but also a negative multiplied by a negative is positive.
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If π₯ plus 0.5 all squared is always positive, then two lots of this number must also be always positive.
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Then we add a positive number.
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We add 0.5.
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A positive plus a positive is always positive.
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And we have shown that our expression two π₯ squared plus two π₯ plus one is always positive.