Verify Equivalence: 5x²+7x+7 = (2x+3)(3x+4)

Question

Are the expressions on both sides equivalent?

5x2+7x+7=?(2x+3)(3x+4) 5x^2+7x+7\stackrel{?}{=}(2x+3)(3x+4)

Video Solution

Step-by-Step Solution

To determine if the expressions are equivalent, we need to expand the right-side expression, (2x+3)(3x+4) (2x + 3)(3x + 4) , and compare it with 5x2+7x+7 5x^2 + 7x + 7 .

Let's expand the right-side expression:

  • First, use the distributive property (or FOIL):
  • (2x+3)(3x+4)=2x3x+2x4+33x+34(2x + 3)(3x + 4) = 2x \cdot 3x + 2x \cdot 4 + 3 \cdot 3x + 3 \cdot 4
  • This simplifies to 6x2+8x+9x+126x^2 + 8x + 9x + 12.
  • Combine like terms: 6x2+(8x+9x)+12=6x2+17x+126x^2 + (8x + 9x) + 12 = 6x^2 + 17x + 12.

Now, compare the expanded expression 6x2+17x+126x^2 + 17x + 12 to the left side 5x2+7x+75x^2 + 7x + 7:

  • The coefficient of x2x^2 is 6 on the right, but 5 on the left.
  • The coefficient of xx is 17 on the right, but 7 on the left.
  • The constant term is 12 on the right, but 7 on the left.

Since all corresponding coefficients differ between the two sides, the expressions are not equivalent.

Therefore, the correct answer is: No, because all the coefficients of the corresponding terms in the expressions on both sides of the equation are different.

Answer

No, because all the coefficients of the corresponding terms in the expressions on both sides of the equation are different.