Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

at^(2)+(7)/(2)t-4=0
The given equation has solutions at
t=-8 and
t=1. What is the value of the constant
a ?

$a t^{2}+\frac{7}{2} t-4=0$ $\newline$ The given equation has solutions at $t=-8$ and $t=1$ . What is the value of the constant $a$ ?

View step-by-step help

View step-by-step help

Write and solve direct variation equations

Full solution

Q.

a t^{2}+\frac{7}{2} t-4=0

\newline

The given equation has solutions at

t=-8

and

t=1

. What is the value of the constant

a

?

Given Quadratic Equation: We are given that the quadratic equation $a t^{2} + \left(\frac{7}{2}\right)t - 4 = 0$ has solutions $t = -8$ and $t = 1$ . We can use these solutions to find the value of the constant $a$ by plugging them into the equation and solving for $a$ . $\newline$ First, let's plug in $t = -8$ .
Substitute $t = -8$ : Substitute $t = -8$ into the equation: $\newline$ $a(-8)^2 + \left(\frac{7}{2}\right)(-8) - 4 = 0$ $\newline$ Simplify the equation: $\newline$ $64a - 28 - 4 = 0$ $\newline$ $64a - 32 = 0$
Solve for $a$ ( $t = -8$ ): Now, solve for $a$ : $64a = 32$ $a = \frac{32}{64}$ $a = \frac{1}{2}$
Substitute $t = 1$ : We have found a value for $a$ using the solution $t = -8$ . However, we must also verify that this value of $a$ works for the other solution, $t = 1$ , to ensure that there is no math error. $\newline$ Substitute $t = 1$ into the original equation: $\newline$ $a(1)^2 + (\frac{7}{2})(1) - 4 = 0$ $\newline$ Simplify the equation: $\newline$ $a + \frac{7}{2} - 4 = 0$
Solve for $a$ ( $t = 1$ ): Now, solve for $a$ using the equation from the previous step: $\newline$ $a = 4 - \frac{7}{2}$ $\newline$ $a = \frac{8}{2} - \frac{7}{2}$ $\newline$ $a = \frac{1}{2}$
Verify Consistency: The value of $a$ is consistent for both solutions $t = -8$ and $t = 1$ . Therefore, the value of the constant $a$ is $\frac{1}{2}$ .

More problems from Write and solve direct variation equations

Question

In a direct variation,

y = 6

x = 2

. Write a direct variation equation that shows the relationship between

x

y

\newline

Write your answer as an equation with

y

first, followed by an equals sign.

Posted 5 months ago

Question

y

varies directly with

x

y = 12

x = 6

y

x = 4

\newline

Write and solve a direct variation equation to find the answer.

\newline

y =

Posted 5 months ago

Question

Which equation shows inverse variation?

\newline

\newline

(\frac{x}{36}) = y

\newline

yx = -74

Posted 5 months ago

Question

y

varies inversely with

x

y = 1

x = 4

y

x = 2

\newline

Write and solve an inverse variation equation to find the answer.

\newline

y =

Posted 5 months ago

Question

What is the value of

\frac{x^{2}}{y^{4}}

x=8

y=2

Posted 5 months ago

Question

(x+4)(2 x-a)=0

\newline

In the given equation,

a

is a constant. If the equation has the solutions

x=4

x=-4

, what is the value of

a

Posted 3 months ago

Question

(2 x+5)(-m x+9)=0

\newline

In the given equation,

m

is a constant. If the equation has the solutions

x=-\frac{5}{2}

x=\frac{3}{2}

, what is the value of

m

Posted 3 months ago

Question

(4 x+3)-(5 x-7)=a x+b

\newline

In the given equation,

b

is a constant. What is the value of

b

Posted 3 months ago

Question

y=\frac{29}{8} x-\frac{2}{3}

\newline

The given equation is graphed in the

x y

-plane. What is the slope of the line?

Posted 8 months ago

Question

You pick a card at random, put it back, and then pick another card at random.

\newline

4

\newline

5

\newline

6

\newline

7

\newline

What is the probability of picking a number greater than

5

and then picking a

4

\newline

Write your answer as a percentage.

Posted 2 months ago

Related topics

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