Find the Missing Symbol: (5+x)² + 3x² ? 5x² + 10x + 25

Video Solution

Solution Steps

00:00 Complete the sign

00:06 We'll use shortened multiplication formulas to open the parentheses

00:20 We'll solve the squares and multiplications

00:39 We'll reduce what we can

00:59 X is less than 1 but still positive

01:11 Therefore even when squared it's still less than 1

01:17 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we'll start by expanding both expressions to compare them.

First, expand the left expression:

$1 + (5 + x)^2 + 3x^2$

Using the formula for expanding a square, $(5 + x)^2$ becomes:

$5^2 + 2 \cdot 5 \cdot x + x^2 = 25 + 10x + x^2$

Thus, the left-hand expression becomes:

$1 + 25 + 10x + x^2 + 3x^2 = 26 + 10x + 4x^2$

Next, simplify the right-hand side:

$5x^2 + 10x + 25$

Now, compare both simplified expressions:

Subtract the right expression from the left to see which is greater:

$(26 + 10x + 4x^2) - (5x^2 + 10x + 25)$

$= 26 + 10x + 4x^2 - 5x^2 - 10x - 25$

$= 1 - x^2$

Since $0 < x < 1$ , it follows that $-1 < -x^2 < 0$ , making $1 - x^2$ positive.

Therefore, the expression on the left is greater than the expression on the right:

$1 + (5 + x)^2 + 3x^2 \gt 5x^2 + 10x + 25$ .