Video: Solving Quadratic Equations by Factoring Perfect Squares

Solve the equation 4𝑡² − 32𝑡 + 64 = 0 by factoring.

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Video Transcript

Solve the equation four 𝑡 squared minus 32 𝑡 plus 64 equals zero by factoring.

The first thing to note here is that all three terms on the left-hand side of the equation are divisible by four. Therefore, we can divide both sides of the equation by four. Four 𝑡 squared divided by four is 𝑡 squared, negative 32 𝑡 divided by four is negative eight 𝑡, and 64 divided by four is 16. On the right-hand side, zero divided by four is still zero.

We then need to factorize the quadratic equation by putting it into brackets or parentheses. As the coefficient of 𝑡 squared is one, the first term in both parentheses will be 𝑡 as 𝑡 multiplied by 𝑡 is 𝑡 squared.

In order to work out the second terms in both of the parentheses, we need to find two numbers that multiply to give us positive 16. Those same two numbers also need to add to give us negative eight. What two numbers multiply to give us positive 16 and add to give negative eight? Well, negative four multiplied by negative four is positive 16 and negative four plus negative four is negative eight. Therefore, the two numbers in the parentheses are negative four and negative four. This gives us 𝑡 minus four multiplied by 𝑡 minus four.

As the two brackets are identical, there will only be one solution to the quadratic equation. This will occur when 𝑡 minus four is equal to zero. Adding four to both sides of this equation gives us an answer of 𝑡 equals four. Therefore, the only solution to the quadratic equation four 𝑡 squared minus 32 𝑡 plus 64 equals zero is 𝑡 equal to four.

We can check this answer by substituting 𝑡 equals four into the original equation four multiplied by four squared minus 32 multiplied by four plus 64. Well, four multiplied by four squared is 64, negative 32 multiplied by four is negative 128, and finally, 64 minus 128 plus 64 equals zero. Therefore, our answer is correct.

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