Simplify the following expressions: (a) (a+2)(a+7), (b) (a+4)(a−6), (c) (x−3)(x−9), (d) (m+5)(m−5), (e)…

Question

Simplify the following expressions: (a) (a+2)(a+7), (b) (a+4)(a−6), (c) (x−3)(x−9), (d) (m+5)(m−5), (e) (y−7)(y+7).

ScanSolve AI Tutor · Accepted Answer

a) Simplify (a+2)(a+7). Step 1: Use the FOIL method (First, Outer, Inner, Last) to multiply the binomials. (a+2)(a+7) = a · a + a · 7 + 2 · a + 2 · 7 Step 2: Perform the multiplications. a² + 7a + 2a + 14 Step 3: Combine like terms. a² + 9a + 14 The simplified expression is a² + 9a + 14. b) Simplify (a+4)(a−6). (Assuming the second bracket is (a−6) based on the image clarity.) Step 1: Use the FOIL method to multiply the binomials. (a+4)(a−6) = a · a + a · (−6) + 4 · a + 4 · (−6) Step 2: Perform the multiplications. a² − 6a + 4a − 24 Step 3: Combine like terms. a² − 2a − 24 The simplified expression is a² − 2a − 24. c) Simplify (x−3)(x−9). Step 1: Use the FOIL method to multiply the binomials. (x−3)(x−9) = x · x + x · (−9) + (−3) · x + (−3) · (−9) Step 2: Perform the multiplications. x² − 9x − 3x + 27 Step 3: Combine like terms. x² − 12x + 27 The simplified expression is x² − 12x + 27. d) Simplify (m+5)(m−5). Step 1: Recognize this as a difference of squares pattern, (A+B)(A−B) = A² − B². (m+5)(m−5) = m² − 5² Step 2: Calculate the square. m² − 25 The simplified expression is m² − 25. e) Simplify (y−7)(y+7). Step 1: Recognize this as a difference of squares pattern, (A−B)(A+B) = A² − B². (y−7)(y+7) = y² − 7² Step 2: Calculate the square. y² − 49 The simplified expression is y² − 49. f) Simplify (2a+7)². Step 1: Use the square of a binomial formula, (A+B)² = A² + 2AB + B². (2a+7)² = (2a)² + 2(2a)(7) + 7² Step 2: Perform the multiplications and squaring. 4a² + 28a + 49 The simplified expression is 4a² + 28a + 49. g) Simplify (2x²−8)². Step 1: Use the square of a binomial formula, (A−B)² = A² − 2AB + B². (2x²−8)² = (2x²)² − 2(2x²)(8) + 8² Step 2: Perform the multiplications and squaring. 4x⁴ − 32x² + 64 The simplified expression is 4x⁴ − 32x² + 64. h) Simplify −2(x−3)² − 3(x+1)(2x−5). Step 1: Expand (x−3)². (x−3)² = x² − 2(x)(3) + 3² = x² − 6x + 9 Step 2: Expand (x+1)(2x−5). (x+1)(2x−5) = x(2x) + x(−5) + 1(2x) + 1(−5) = 2x² − 5x + 2x − 5 = 2x² − 3x − 5 Step 3: Substitute the expanded forms back into the original expression. −2(x² − 6x + 9) − 3(2x² − 3x − 5) Step 4: Distribute the constants. −2x² + 12x − 18 − 6x² + 9x + 15 Step 5: Combine like terms. (−2x² − 6x²) + (12x + 9x) + (−18 + 15) −8x² + 21x − 3 The simplified expression is −8x² + 21x − 3. i) Simplify −2(m−4)² − 3m − (m+6)(m−6). Step 1: Expand (m−4)². (m−4)² = m² − 2(m)(4) + 4² = m² − 8m + 16 Step 2: Expand (m+6)(m−6) using the difference of squares formula. (m+6)(m−6) = m² − 6² = m² − 36 Step 3: Substitute the expanded forms back into the original expression. −2(m² − 8m + 16) − 3m − (m² − 36) Step 4: Distribute the constants and the negative sign. −2m² + 16m − 32 − 3m − m² + 36 Step 5: Combine like terms. (−2m² − m²) + (16m − 3m) + (−32 + 36) −3m² + 13m + 4 The simplified expression is −3m² + 13m + 4.

Simplify the following expressions: (a) (a+2)(a+7), (b) (a+4)(a-6), (c) (x-3)(x-9), (d) (m+5)(m-5), (e) (y-7)(y+7).

Step-by-step solution

Part a· 3 steps

Use the FOIL method (First, Outer, Inner, Last) to multiply the binomials.

Perform the multiplications.

Combine like terms.

Part b· 3 steps

Use the FOIL method to multiply the binomials.

Perform the multiplications.

Combine like terms.

Part c· 3 steps

Use the FOIL method to multiply the binomials.

Perform the multiplications.

Combine like terms.

Part d· 2 steps

Recognize this as a difference of squares pattern, $(A+B)(A-B) = A^2 - B^2$ .

Calculate the square.

Part e· 2 steps

Recognize this as a difference of squares pattern, $(A-B)(A+B) = A^2 - B^2$ .

Calculate the square.

Part f· 2 steps

Use the square of a binomial formula, $(A+B)^2 = A^2 + 2AB + B^2$ .

Perform the multiplications and squaring.

Part g· 2 steps

Use the square of a binomial formula, $(A-B)^2 = A^2 - 2AB + B^2$ .

Perform the multiplications and squaring.

Part h· 5 steps

Expand $(x-3)^2$ .

Expand $(x+1)(2x-5)$ .

Substitute the expanded forms back into the original expression.

Distribute the constants.

Combine like terms.

Part h(i)· 5 steps

Expand $(m-4)^2$ .

Expand $(m+6)(m-6)$ using the difference of squares formula.

Substitute the expanded forms back into the original expression.

Distribute the constants and the negative sign.

Combine like terms.

Still stuck on this one?

More Mathematics Questions

Simplify the following expressions: (a) (a+2)(a+7), (b) (a+4)(a-6), (c) (x-3)(x-9), (d) (m+5)(m-5), (e) (y-7)(y+7).

Step-by-step solution

Part a· 3 steps

Use the FOIL method (First, Outer, Inner, Last) to multiply the binomials.

Perform the multiplications.

Combine like terms.

Part b· 3 steps

Use the FOIL method to multiply the binomials.

Perform the multiplications.

Combine like terms.

Part c· 3 steps

Use the FOIL method to multiply the binomials.

Perform the multiplications.

Combine like terms.

Part d· 2 steps

Recognize this as a difference of squares pattern, (A+B)(A−B)=A2−B2(A+B)(A-B) = A^2 - B^2(A+B)(A−B)=A2−B2.

Calculate the square.

Part e· 2 steps

Recognize this as a difference of squares pattern, (A−B)(A+B)=A2−B2(A-B)(A+B) = A^2 - B^2(A−B)(A+B)=A2−B2.

Calculate the square.

Part f· 2 steps

Use the square of a binomial formula, (A+B)2=A2+2AB+B2(A+B)^2 = A^2 + 2AB + B^2(A+B)2=A2+2AB+B2.

Perform the multiplications and squaring.

Part g· 2 steps

Use the square of a binomial formula, (A−B)2=A2−2AB+B2(A-B)^2 = A^2 - 2AB + B^2(A−B)2=A2−2AB+B2.

Perform the multiplications and squaring.

Part h· 5 steps

Expand (x−3)2(x-3)^2(x−3)2.

Expand (x+1)(2x−5)(x+1)(2x-5)(x+1)(2x−5).

Substitute the expanded forms back into the original expression.

Distribute the constants.

Combine like terms.

Part h(i)· 5 steps

Expand (m−4)2(m-4)^2(m−4)2.

Expand (m+6)(m−6)(m+6)(m-6)(m+6)(m−6) using the difference of squares formula.

Substitute the expanded forms back into the original expression.

Distribute the constants and the negative sign.

Combine like terms.

Still stuck on this one?

More Mathematics Questions

Recognize this as a difference of squares pattern, $(A+B)(A-B) = A^2 - B^2$ .

Recognize this as a difference of squares pattern, $(A-B)(A+B) = A^2 - B^2$ .

Use the square of a binomial formula, $(A+B)^2 = A^2 + 2AB + B^2$ .

Use the square of a binomial formula, $(A-B)^2 = A^2 - 2AB + B^2$ .

Expand $(x-3)^2$ .

Expand $(x+1)(2x-5)$ .

Expand $(m-4)^2$ .

Expand $(m+6)(m-6)$ using the difference of squares formula.