Supply the numbers to write x^2 +10x+16=0

Question

Supply the numbers to write  x^2 +10x+16=0
  in its factored form. Enter the lesser number first.(1 point)
(x+___)
(x+___)=0

GPT-4o mini · Accepted Answer

To factor the quadratic equation $x^2 + 10x + 16 = 0$, we need to find two numbers that multiply to $16$ (the constant term) and add up to $10$ (the coefficient of the linear term). The numbers that satisfy these conditions are $2$ and $8$ because: - $2 \times 8 = 16$ - $2 + 8 = 10$ Thus, we can write the equation in its factored form as: $$ (x + 2)(x + 8) = 0 $$ So, in the blanks: $(x + 2)$ $(x + 8)$ The answer is: **(x + 2)(x + 8) = 0**

Answer

Supply the number so that the quadratic equation below has 6 and 8 as its solutions.(1 point) x^2 −14x+___=0

GPT-4o mini · Answer

To find the missing constant in the quadratic equation $x^2 - 14x + ___ = 0$ such that the solutions (roots) are $6$ and $8$, we can use the fact that for a quadratic equation in the form $x^2 - (sum\ of\ roots)x + (product\ of\ roots) = 0$:

The sum of the roots $6 + 8 = 14$.
The product of the roots $6 \times 8 = 48$.

Thus, we need to fill in the blank with the product of the roots, which is $48$.

So, the equation becomes:

\[ x^2 - 14x + 48 = 0 \]

The missing number is 48.

Answer

Solve the following quadratic equation using the perfect square trinomial pattern: x^2 +10x+25=0 .(1 point) x=___

GPT-4o mini · Answer

The quadratic equation given is $$ x^2 + 10x + 25 = 0. $$ We can recognize that this is a perfect square trinomial, which can be factored as: $$ (x + 5)^2 = 0. $$ Next, we will solve for $x$ by taking the square root of both sides: $$ x + 5 = 0. $$ Now, subtract $5$ from both sides: $$ x = -5. $$ So the solution to the equation is: $$ x = -5. $$ Thus, the answer is **-5**.

Answer

Solve the following quadratic equation using the perfect square trinomial pattern: x^2 +54x=−729 .(1 point) x=___

GPT-4o mini · Answer

To solve the quadratic equation $x^2 + 54x = -729$ using the perfect square trinomial pattern, we first rewrite it in standard form by adding $729$ to both sides: $$ x^2 + 54x + 729 = 0. $$ Next, we want to express the left-hand side as a perfect square trinomial. We can identify that the perfect square trinomial can be formed from: $$ (x + 27)^2 $$ because: $$ (x + 27)(x + 27) = x^2 + 54x + 729. $$ Thus, we rewrite the equation as: $$ (x + 27)^2 = 0. $$ Now, we take the square root of both sides: $$ x + 27 = 0. $$ Next, we solve for $x$ by subtracting $27$ from both sides: $$ x = -27. $$ So the solution to the equation is: $$ x = -27. $$ Thus, the answer is **-27**.

Answer

What are the solutions to x^ 2−121=0 ? Enter the lesser number first.(1 point) x=___ , x=___

GPT-4o mini · Answer

To solve the equation $ x^2 - 121 = 0 $, we can first rewrite it as: $$ x^2 = 121. $$ Next, we take the square root of both sides: $$ x = \pm\sqrt{121}. $$ Since $\sqrt{121} = 11$, we have: $$ x = 11 \quad \text{or} \quad x = -11. $$ Thus, the solutions are: $$ x = -11, \quad x = 11. $$ So, entering the lesser number first, the final answer is: **x = -11, x = 11**.