Factoring Differences of Squares and Perfect Square Trinomials Practice

Confirm there are exactly two terms, subtraction, and two perfect squares.

Both terms are squares: \(1x^2=(1x)^2\) and \(9=3^2\). Therefore \(a^2-b^2=(a+b)(a-b)\) gives the stated factors.

Confirm there are exactly two terms, subtraction, and two perfect squares.

Both terms are squares: \(4x^2=(2x)^2\) and \(25=5^2\). Therefore \(a^2-b^2=(a+b)(a-b)\) gives the stated factors.

Confirm there are exactly two terms, subtraction, and two perfect squares.

Both terms are squares: \(9x^2=(3x)^2\) and \(49=7^2\). Therefore \(a^2-b^2=(a+b)(a-b)\) gives the stated factors.

Confirm there are exactly two terms, subtraction, and two perfect squares.

Both terms are squares: \(16x^2=(4x)^2\) and \(81=9^2\). Therefore \(a^2-b^2=(a+b)(a-b)\) gives the stated factors.

Confirm there are exactly two terms, subtraction, and two perfect squares.

Both terms are squares: \(25x^2=(5x)^2\) and \(121=11^2\). Therefore \(a^2-b^2=(a+b)(a-b)\) gives the stated factors.

Treat each complete squared expression as one base before applying the identity.

Apply \(u^2-v^2=(u+v)(u-v)\) with \(u=m\) and \(v=3\). The conjugate middle terms cancel when expanded.

Treat each complete squared expression as one base before applying the identity.

Apply \(u^2-v^2=(u+v)(u-v)\) with \(u=2p\) and \(v=7\). The conjugate middle terms cancel when expanded.

Treat each complete squared expression as one base before applying the identity.

Apply \(u^2-v^2=(u+v)(u-v)\) with \(u=5r\) and \(v=4\). The conjugate middle terms cancel when expanded.

Treat each complete squared expression as one base before applying the identity.

Apply \(u^2-v^2=(u+v)(u-v)\) with \(u=3a^2\) and \(v=8\). The conjugate middle terms cancel when expanded.

Treat each complete squared expression as one base before applying the identity.

Apply \(u^2-v^2=(u+v)(u-v)\) with \(u=6y^3\) and \(v=11\). The conjugate middle terms cancel when expanded.

After identifying the square bases, verify that twice their product equals the middle term.

The outer terms are \(( 1x )^2\) and \(3^2\), and the middle term is \(2(1x)(3)=6x\). Thus the factorization is \((1x+3)^2\).

After identifying the square bases, verify that twice their product equals the middle term.

The outer terms are \(( 2x )^2\) and \(5^2\), and the middle term is \(2(2x)(5)=20x\). Thus the factorization is \((2x+5)^2\).

After identifying the square bases, verify that twice their product equals the middle term.

The outer terms are \(( 3x )^2\) and \(4^2\), and the middle term is \(2(3x)(4)=24x\). Thus the factorization is \((3x+4)^2\).

After identifying the square bases, verify that twice their product equals the middle term.

The outer terms are \(( 4x )^2\) and \(5^2\), and the middle term is \(2(4x)(5)=40x\). Thus the factorization is \((4x+5)^2\).

After identifying the square bases, verify that twice their product equals the middle term.

The outer terms are \(( 5x )^2\) and \(7^2\), and the middle term is \(2(5x)(7)=70x\). Thus the factorization is \((5x+7)^2\).

The last term remains positive in both perfect-square identities; the middle sign chooses the binomial sign.

The middle coefficient satisfies \(-2(1)(4)=-8\), so the trinomial matches \(u^2-2uv+v^2=(u-v)^2\).

The last term remains positive in both perfect-square identities; the middle sign chooses the binomial sign.

The middle coefficient satisfies \(-2(2)(7)=-28\), so the trinomial matches \(u^2-2uv+v^2=(u-v)^2\).

The last term remains positive in both perfect-square identities; the middle sign chooses the binomial sign.

The middle coefficient satisfies \(-2(3)(5)=-30\), so the trinomial matches \(u^2-2uv+v^2=(u-v)^2\).

The last term remains positive in both perfect-square identities; the middle sign chooses the binomial sign.

The middle coefficient satisfies \(-2(4)(3)=-24\), so the trinomial matches \(u^2-2uv+v^2=(u-v)^2\).

The last term remains positive in both perfect-square identities; the middle sign chooses the binomial sign.

The middle coefficient satisfies \(-2(5)(6)=-60\), so the trinomial matches \(u^2-2uv+v^2=(u-v)^2\).

After extracting a GCF, inspect the remaining expression for another factorable pattern.

First extract the GCF \(6\), leaving \(1x^2-16=(1x)^2-4^2\). Apply the difference-of-squares identity to finish.

After extracting a GCF, inspect the remaining expression for another factorable pattern.

First extract the GCF \(8\), leaving \(9x^2-25=(3x)^2-5^2\). Apply the difference-of-squares identity to finish.

After extracting a GCF, inspect the remaining expression for another factorable pattern.

First extract the GCF \(10\), leaving \(4x^2-49=(2x)^2-7^2\). Apply the difference-of-squares identity to finish.

After extracting a GCF, inspect the remaining expression for another factorable pattern.

First extract the GCF \(12\), leaving \(25x^2-36=(5x)^2-6^2\). Apply the difference-of-squares identity to finish.

After extracting a GCF, inspect the remaining expression for another factorable pattern.

First extract the GCF \(14\), leaving \(49x^2-64=(7x)^2-8^2\). Apply the difference-of-squares identity to finish.

Set each complete factor equal to zero and retain both branches.

Factor as \(( 1x+6 )( 1x-6 )=0\). The Zero Product Property gives both roots \(x=\pm6\).

Set each complete factor equal to zero and retain both branches.

Factor as \(( 2x+9 )( 2x-9 )=0\). The Zero Product Property gives both roots \(x=\pm4.5\).

Set each complete factor equal to zero and retain both branches.

Factor as \(( 3x+10 )( 3x-10 )=0\). The Zero Product Property gives both roots \(x=\pm3.3333333333333335\).

Set each complete factor equal to zero and retain both branches.

Factor as \(( 4x+7 )( 4x-7 )=0\). The Zero Product Property gives both roots \(x=\pm1.75\).

Set each complete factor equal to zero and retain both branches.

Factor as \(( 5x+12 )( 5x-12 )=0\). The Zero Product Property gives both roots \(x=\pm2.4\).

Recognize the perfect square before applying the Zero Product Property.

The trinomial is \((1x-3)^2\). A square equals zero only when its base equals zero, giving \(x=3\) as a repeated root.

Recognize the perfect square before applying the Zero Product Property.

The trinomial is \((2x+5)^2\). A square equals zero only when its base equals zero, giving \(x=-2.5\) as a repeated root.

Recognize the perfect square before applying the Zero Product Property.

The trinomial is \((3x-4)^2\). A square equals zero only when its base equals zero, giving \(x=1.3333333333333333\) as a repeated root.

Recognize the perfect square before applying the Zero Product Property.

The trinomial is \((4x+7)^2\). A square equals zero only when its base equals zero, giving \(x=-1.75\) as a repeated root.

Recognize the perfect square before applying the Zero Product Property.

The trinomial is \((5x-2)^2\). A square equals zero only when its base equals zero, giving \(x=0.4\) as a repeated root.

Match the given sum and difference directly to the conjugate factors.

Use \(u^2-v^2=(u+v)(u-v)\). Substitution gives \(14(6)=84\) without solving separately for both variables.

Match the given sum and difference directly to the conjugate factors.

Use \(u^2-v^2=(u+v)(u-v)\). Substitution gives \(19(5)=95\) without solving separately for both variables.

Match the given sum and difference directly to the conjugate factors.

Use \(u^2-v^2=(u+v)(u-v)\). Substitution gives \(12(-4)=-48\) without solving separately for both variables.

Match the given sum and difference directly to the conjugate factors.

Use \(u^2-v^2=(u+v)(u-v)\). Substitution gives \(-9(7)=-63\) without solving separately for both variables.

Match the given sum and difference directly to the conjugate factors.

Use \(u^2-v^2=(u+v)(u-v)\). Substitution gives \(25(3)=75\) without solving separately for both variables.

Recognition conditions are stricter than merely seeing square terms.

Counting terms, checking the operation sign, identifying square outer terms, and testing the middle term leads to: perfect square with a positive middle term.

Recognition conditions are stricter than merely seeing square terms.

Counting terms, checking the operation sign, identifying square outer terms, and testing the middle term leads to: perfect square with a negative middle term.

Recognition conditions are stricter than merely seeing square terms.

Counting terms, checking the operation sign, identifying square outer terms, and testing the middle term leads to: difference of squares.

Recognition conditions are stricter than merely seeing square terms.

Counting terms, checking the operation sign, identifying square outer terms, and testing the middle term leads to: none of these real factoring patterns.

Recognition conditions are stricter than merely seeing square terms.

Counting terms, checking the operation sign, identifying square outer terms, and testing the middle term leads to: perfect square with a positive middle term.

Use structural conditions and expansion rather than visual resemblance.

The proposed square creates a nonzero middle term. Expanding a proposed identity or checking the Zero Product Property's hypothesis verifies the diagnosis.

Use structural conditions and expansion rather than visual resemblance.

The expression is a sum, not a difference, of squares. Expanding a proposed identity or checking the Zero Product Property's hypothesis verifies the diagnosis.

Use structural conditions and expansion rather than visual resemblance.

The middle term does not equal twice the product of the square bases. Expanding a proposed identity or checking the Zero Product Property's hypothesis verifies the diagnosis.

Use structural conditions and expansion rather than visual resemblance.

The product must equal zero before individual factors can be set to zero. Expanding a proposed identity or checking the Zero Product Property's hypothesis verifies the diagnosis.

Use structural conditions and expansion rather than visual resemblance.

Each factor can equal zero, so both resulting roots must be considered. Expanding a proposed identity or checking the Zero Product Property's hypothesis verifies the diagnosis.