Find all values of b or c that will make the polynomial a perfect square trinomial. 49x²+70x+c

검증된 단계별 안내

Recognize that a perfect square trinomial takes the form $\left( dx + e \right)^2 = d^2x^2 + 2dex + e^2$. Our goal is to express $49x^2 + 70x + c$ in this form.

Identify the coefficient of $x^2$ in the given polynomial, which is 49. Since $d^2 = 49$, solve for $d$ by taking the square root: $d = \pm 7$.

Use the middle term \$70x$ to find $e$. Recall that the middle term in the perfect square trinomial is \$2de x$. Substitute $d = 7$ (choosing the positive root for simplicity) and set $2 \times 7 \times e = 70$.

Solve for $e$ from the equation $14e = 70$, which gives $e = 5$.

Find $c$ by calculating $e^2$, since the constant term in the perfect square trinomial is $e^2$. Therefore, $c = 5^2 = 25$.

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Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, typically in the form (mx + n)^2. It expands to m^2x^2 + 2mnx + n^2, where the middle term is twice the product of the square roots of the first and last terms.

Identifying Coefficients in a Quadratic

In a quadratic expression ax^2 + bx + c, the coefficients a, b, and c represent the quadratic, linear, and constant terms respectively. Understanding their roles helps in comparing the given polynomial to the standard form of a perfect square trinomial to find missing values.

Using the Relationship Between Coefficients for Perfect Squares

For a trinomial to be a perfect square, the constant term c must equal (b/2)^2 when the leading coefficient a is 1, or more generally, c = (b/(2a))^2 * a. This relationship allows solving for unknown coefficients to ensure the polynomial is a perfect square.

Find all values of b or c that will make the polynomial a perfect square trinomial. 49x2+70x+c