The Perfect Square Trinomial Calculator, provided by Hesapstan, checks whether an integer-coefficient trinomial ax²+bx+c can be written as (px+q)² or (px−q)². It also builds a perfect square trinomial from p, q, and a sign choice, showing p²x², ±2pqx, and q² step by step. This is an identity and factoring tool; it does not solve equations or compute roots.
A perfect square trinomial is the expansion of a squared binomial
A perfect square trinomial comes from expanding (px+q)² or (px−q)². That means the x² coefficient must be p², the constant term must be q², and the middle term must be +2pqx or −2pqx.
This calculator has two modes. Check / Factor mode tests a given ax²+bx+c and shows the factored form when it is a PST. Build mode starts from p, q, and a sign and expands the squared binomial.
A number perfect-square calculator checks values such as 49 or 121. This tool checks trinomials containing x, such as x²+6x+9 or 4x²−12x+9.
Check mode compares the square roots and the middle term
In Check mode, a, b, and c must be integers, with a not equal to zero. The runtime checks whether a>0 and c>0, whether sqrt(a) and sqrt(c) are integers, and whether the absolute value of b equals 2pq.
- For 4x²+12x+9, take a=4, b=12, and c=9.
- sqrt(4)=2 and sqrt(9)=3, so p=2 and q=3.
- The expected middle coefficient is 2·2·3=12.
- b is positive and |b|=12, so the trinomial is a perfect square trinomial.
- The factored form is (2x+3)².
The input may be valid, but the (px±q)² form requires positive square values for the x² coefficient and constant term. If a<0 or c<0, this tool reports Not PST.
The expected middle term explains near misses
A trinomial can look close to a perfect square but fail because the middle term is wrong. When the first and last terms are usable squares, the calculator shows the middle term that would be required and compares it with the actual b·x term.
For example, x²+0x+9 has square-looking first and last terms. But p=1 and q=3 require a middle term of 2·1·3x=6x. The actual middle term is 0x, so the expression is not a perfect square trinomial.
If b is positive, the form is (px+q)². If b is negative, the form is (px−q)². For instance, 4x²−12x+9 factors as (2x−3)².
Build mode creates the identity from p and q
Build mode takes integer p and q values and a separate sign choice. p cannot be zero. The calculator then expands the selected form into p²x², ±2pqx, and q².
- Choose p=3, q=2, and the plus sign.
- The factored form is (3x+2)².
- The first term is p²x²=9x².
- The middle term is +2pqx=+12x.
- The last term is q²=4.
- The expanded result is 9x²+12x+4.
With the minus sign, only the middle term changes sign. The expansion of (3x−2)² is 9x²−12x+4; the first and last terms remain positive.
No GCF is extracted before the PST check
The calculator checks the trinomial exactly as entered. If all terms share a greatest common factor, the tool does not first factor that GCF out and then test the remaining trinomial.
For example, 2x²+8x+8 can be rewritten manually as 2(x²+4x+4)=2(x+2)². But the runtime sees a=2 directly, and 2 is not a perfect square, so it does not label the entered trinomial as a PST.
If a common factor is present, removing it is a separate algebra step. This calculator does not promise that preprocessing step.
This differs from numeric binomial squares and completing the square
This calculator works with polynomial coefficients and the form ax²+bx+c. A numeric square-of-a-binomial tool computes values such as (a+b)² using numbers only. Completing the square is different again: it transforms and solves a quadratic equation.
- Square of a Binomial: numeric (a±b)² calculation.
- Perfect Square Trinomial: checks whether ax²+bx+c equals (px±q)².
- Completing the Square: solves or transforms ax²+bx+c=0.
- Discriminant: interprets root type, not the PST identity.
The input is the expression ax²+bx+c. Roots, graph behavior, and equation solving are outside this calculator's output.
Frequently Asked Questions
How do I know whether a trinomial is a perfect square trinomial?
a and c must be positive perfect squares. If sqrt(a)=p and sqrt(c)=q, the absolute value of b must equal 2pq. The sign of b determines whether the form is (px+q)² or (px−q)².
What happens if a GCF should be factored out first?
This calculator does not extract a GCF before checking. A trinomial such as 2x²+8x+8 may contain a PST after factoring out 2, but the entered trinomial is not labeled as PST by this runtime.
What if b is negative?
If |b| matches the expected 2pq value and b is negative, the factored form uses a minus sign: (px−q)².
What does expected middle term mean?
It is the middle term that a perfect square trinomial would need after p and q are determined from the first and last terms. The calculator compares it with the actual b·x term.
Does this calculator solve quadratic equations?
No. It checks and builds a polynomial identity. Use a completing-the-square or quadratic-solving calculator for equation roots.