Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

The given equation $a imes x^2 + 98x + c$ has at least $1$ real root and a factor of $kx + j$ . What is the greatest possible value of $ac$ ?

View step-by-step help

View step-by-step help

Simplify radical expressions with variables II

Full solution

Q. The given equation

a imes x^2 + 98x + c

has at least

1

real root and a factor of

kx + j

. What is the greatest possible value of

ac

?

Discriminant Calculation: Since the equation has at least one real root, the discriminant must be non-negative. $\newline$ Discriminant: $b^2 - 4ac \geq 0$ $\newline$ Here, $b = 98$ , so: $\newline$ $98^2 - 4ac \geq 0$
Calculate $98$ ^ $2$ : Calculate $98^2$ : $\newline$ $98^2 = 9604$ $\newline$ So: $\newline$ $9604 - 4ac \geq 0$
Solve for ac: Rearrange to solve for $ac$ : $\newline$ $9604 \geq 4ac$ $\newline$ Divide both sides by $4$ : $\newline$ $2401 \geq ac$
Factorize Polynomial: Since $kx + j$ is a factor, the polynomial can be written as: $\newline$ $a(x - r)(x - s)$ $\newline$ Expanding: $\newline$ $a(x^2 - (r+s)x + rs)$ $\newline$ Comparing with $a \cdot x^2 + 98x + c$ : $\newline$ $-a(r+s) = 98$ and $ars = c$
Maximize ac: To maximize $ac$ , we need to maximize $rs$ while keeping $-a(r+s) = 98$ . $\newline$ Let $r = s$ for simplicity: $\newline$ $-2ar = 98$ $\newline$ So: $\newline$ $ar = -49$
Substitute ar: Substitute $ar = -49$ into $ars = c$ : $\newline$ $a(-49)r = c$ $\newline$ So: $\newline$ $c = -49r$
Final Calculation: Now, $ac = a \cdot (-49r)$ : $\newline$ $ac = -49ar$ $\newline$ Since $ar = -49$ : $\newline$ $ac = -49(-49)$ $\newline$ $ac = 2401$

More problems from Simplify radical expressions with variables II

Question

Find the real-number root.

\newline

\sqrt{16}

\newline

\newline

Write your answer in simplified form.

\newline

\_\_\_\_

Posted 6 months ago

Question

Find the real-number root.

\newline

\sqrt{\frac{9}{16}}

\newline

Simplify your answer and write it as a decimal or as a proper or improper fraction.

\newline

\_\_\_\_\_

Posted 6 months ago

Question

Find the real-number root using a calculator. Round your answer to the nearest thousandth.

\newline

\newline

Posted 6 months ago

Question

Multiply. Write your answer in simplest form.

\newline

\sqrt{273} \times \sqrt{42}

\newline

Posted 5 months ago

Question

\newline

\sqrt{\frac{63}{4}}

\newline

Posted 6 months ago

Question

Simplify. Rationalize the denominator.

\newline

\frac{-4}{9 - \sqrt{2}}

\newline

Posted 6 months ago

Question

z

\newline

16 = \sqrt{z}

\newline

z =

Posted 6 months ago

Question

Simplify the radical expression.

\sqrt{14x^{14}}

Write your answer in the form

A

\sqrt{B}

A\sqrt{B}

A

B

are constants or expressions in

x

. Use at most one radical in your answer, and at most one absolute value symbol in your expression for

A

Posted 6 months ago

Question

Simplify the radical expression.

\newline

\sqrt{12x^{12}}

\newline

Write your answer in the form

A

\sqrt{B}

A\sqrt{B}

A

B

are constants or expressions in

x

. Use at most one radical in your answer, and at most one absolute value symbol in your expression for

A

\newline

\underline{\hspace{3cm}}

Posted 6 months ago

Question

What is the value of

\sin \left(\frac{5 \pi}{6}\right) ?

\newline

1

\newline

-\frac{\sqrt{3}}{2}

\newline

-\frac{\sqrt{2}}{2}

\newline

\frac{1}{2}

\newline

150

Posted 8 months ago

Related topics

Algebra - Order of Operations Algebra - Distributive Property `X` and `Y` Axes Geometry - Scalene Triangle Common Multiple Geometry - Quadrant