# Solve Quadratic Equation: Find c-a Given p and q

• MHB
• Monoxdifly
In summary, the value of c-a is 10-4c, where c is determined by the quadratic equation a^2+9a-90=0 and a=6.
Monoxdifly
MHB
Given p and q are the roots of the quadratic equation $$\displaystyle ax^2-5x+c=0$$ with $$\displaystyle a\neq0$$. If $$\displaystyle p,q,\frac1{8pq}$$ forms a geometric sequence and $$\displaystyle log_a18+log_ap=1$$, the value of c – a is ...
A. $$\displaystyle \frac13$$
B. $$\displaystyle \frac12$$
C. 3
D. 5
E. 7

Since $$\displaystyle p,q,\frac1{8pq}$$ is a geometric sequence, then:
$$\displaystyle \frac{q}{p}=\frac{\frac1{8pq}}q$$
$$\displaystyle \frac{q}{p}=\frac1{8pq^2}$$
$$\displaystyle q=\frac1{8q^2}$$
$$\displaystyle q^3=\frac18$$
$$\displaystyle q=\frac12$$

Also, since $$\displaystyle log_a18+log_ap=1$$, then:
$$\displaystyle log_a18p=log_aa$$
18p = a
$$\displaystyle p=\frac{a}{18}$$

This is where the real problem starts. No matter how I substitute, either it will cancel out the a's or p's, or becoming a quadratic equation with no real roots. What should I do?

Additionally we have $a(x-p)(x-q)=ax^2 - a(p+q)x + apq = ax^2-5x+c$.
So $a(p+q)=5$ and $apq = c$.
If we substitute the $p$ and $q$ that we've found, we can find $a$ and an expression for $c$ in $a$, and finally $c-a$.

a(a/18)(1/2) = c
(a^2)/36 = c
a^2 = 36c

a(p + q) = 5
a(a/18 + 1/2) = 5
(a^2)/18 + 1/2 a = 5
36c/18 + 1/2 a = 5
2c + 1/2 a = 5
1/2 a = 5 - 2c
a = 10 - 4c
c - a = c - (10 - 4c) = -3c - 10
Sorry, still stuck.

$a(p+q) = 5 \implies a\left(\dfrac{a}{18}+\dfrac{1}{2}\right) = 5 \implies a^2+9a-90=0 \implies a = 6$ since $a>0$ (why?)

finally, you have $c = apq$

you should be able to finish from here

btw ... $c-(10-4c) \ne -3c-10$

Ah, I see. Thank you very much! :D

## 1. What is a quadratic equation?

A quadratic equation is an equation in the form of ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. It is a type of polynomial equation and has the highest degree of 2.

## 2. How do I solve a quadratic equation?

To solve a quadratic equation, you can use the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / 2a. You can also factor the equation or complete the square to find the solutions.

## 3. What is the purpose of finding c-a in a quadratic equation?

In a quadratic equation, c-a is the constant term, which represents the y-intercept of the parabola. Finding c-a allows you to graph the equation and understand its behavior.

## 4. What are p and q in a quadratic equation?

In a quadratic equation, p and q are the coefficients of the x-term and the constant term, respectively. They are used to plug into the quadratic formula or to factor the equation.

## 5. Can a quadratic equation have more than one solution?

Yes, a quadratic equation can have two solutions, one solution, or no real solutions, depending on the discriminant (b^2 - 4ac). If the discriminant is positive, there are two real solutions. If it is zero, there is one real solution. If it is negative, there are no real solutions.

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