Operations & Word Problems

Depth · one concept from first principles · step 2 of 4

Linear means the square term's coefficient is zero

When the expression below is simplified, find the value of the constant aa that makes it a linear expression in xx, and find that linear expression.

2a2x2−3x+2−2x2+ax+4x−32a^2x^2-3x+2-2x^2+ax+4x-3

AExpressions
Your answer
How to solve
Strategy Identify Subproblems — Collect like terms by degree so each coefficient is visible. A linear expression needs a zero x² coefficient and a nonzero x coefficient; the first gives two candidates for a, and the second rules one out.
1STEP 1

Group by degree

Collect the x² terms, the x terms and the constants.

(2a²-2)x²+(-3+a+4)x+(2-3)=(2a²-2)x²+(a+1)x-1
2STEP 2

Make the square term vanish

For the expression to be linear, the x² coefficient must be 0. A number whose square is 1 is 1 or -1, since (-1)×(-1)=1.

2a²-2=0 → a²=1 → a=1 or a=-1
3STEP 3

Keep the x term alive

The x coefficient a+1 must not be 0. With a=-1 it is 0, leaving just the constant -1, which is not linear, so a=-1 is out.

a+1 ≠ 0 → a ≠ -1 → a=1
4STEP 4

Write the linear expression

Put a=1 into the simplified form.

(2-2)x²+(1+1)x-1=2x-1
Answer
a=1, 2x-1
Substitute a=1 into the original: 2x²-3x+2-2x²+x+4x-3=2x-1. With a=-1 you get 2x²-3x+2-2x²-x+4x-3=-1, a constant, confirming that a=-1 fails.
Takeaway

Linear means the x² part is gone but the x part is still there; check both conditions.

  • Group by degree
  • Make the square term vanish
  • Keep the x term alive
  • Write the linear expression
Where next?
Level upsuggested
No solution means 0 times x equals a nonzero numberMatch coefficients of like terms
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
Sum pyramid: symmetric terms cancel at the top

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▶ Practice — 12 problems