Solve the Quadratic Equation for X: ax + a^2 = (a + x)^2 + 3

Look at the equation below and express X in terms of a, given that a<3.

$ax+a^2=(a+x)^2+3$

Video Solution

Solution Steps

00:00 Express X in terms of A

00:04 Expand brackets properly according to shortened multiplication formulas

00:15 Arrange the equation so that one side equals 0

00:23 Arrange the equation

00:26 Look at the coefficients (every number is actually multiplied by 1)

00:32 Use the quadratic formula to find possible solutions

00:43 Substitute appropriate values and solve to find the solutions

01:06 A is less than 3 according to the given data

01:09 Therefore, A squared is definitely less than 9

01:13 Thus, the above root expression is not logical - less than 0

01:19 Therefore, there is no solution

Step-by-Step Solution

To find $x$ in terms of $a$ from the equation $ax + a^2 = (a + x)^2 + 3$ , follow these steps:

Step 1: Expand the right-hand side:
$(a + x)^2 = a^2 + 2ax + x^2$
Step 2: Substitute and simplify the equation:
$ax + a^2 = a^2 + 2ax + x^2 + 3$
Step 3: Rearrange the terms:
$ax + a^2 - a^2 = 2ax + x^2 + 3$
$ax = 2ax + x^2 + 3$
Step 4: Move all terms to one side of the equation to form a quadratic:
$0 = x^2 + ax + 3$
Step 5: Analyze the quadratic equation $x^2 + ax + 3 = 0$ :
Calculate the discriminant: $\Delta = a^2 - 4 \cdot 1 \cdot 3 = a^2 - 12$ .
Step 6: Consider the condition for real solutions ( $\Delta \geq 0$ ):
Since $a^2 - 12 < 0$ for $a < 3$ , the discriminant is negative for values of $a < 3$ . This means the quadratic equation has no real solutions.

Therefore, for the given condition $a < 3$ , there is no solution for $x$ .