Problem 50 - Algebra and Trigonometry

Question

Find all real solutions of the equation. $$ 25 x^{2}+70 x+49=0 $$

Step-by-Step Solution

Answer

The real solution is \( x = -\frac{7}{5} \).

Step 1: Identify the Quadratic Equation

The given expression is \( 25x^2 + 70x + 49 = 0 \), which is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), where \( a = 25 \), \( b = 70 \), and \( c = 49 \).

Step 2: Check for Perfect Square Trinomial

A quadratic expression can sometimes be factored as a perfect square; \( a^2 + 2ab + b^2 = (a+b)^2 \). Here, check if \( 25x^2 + 70x + 49 \) can be written in this form. Look for terms: \( (5x)^2 \), \( 2 \times 5x \times 7 \), and \( 7^2 \).

Step 3: Rewrite as a Perfect Square

Recognize that \( 25x^2 + 70x + 49 = (5x + 7)^2 \). This shows the quadratic is a perfect square trinomial.

Step 4: Solve for x using the Square

Since \( (5x + 7)^2 = 0 \), take the square root of both sides: \( 5x + 7 = 0 \).

Step 5: Isolate x

Rearrange \( 5x + 7 = 0 \) to find \( x \). Subtract 7 from both sides to get \( 5x = -7 \).

Step 6: Solve for x

Divide both sides by 5 to solve for x: \( x = -\frac{7}{5} \).

Key Concepts

Perfect Square Trinomial

Quadratic equations can sometimes be expressed as perfect square trinomials. A perfect square trinomial is a quadratic expression that can be written as \((a+b)^2=a^2+2ab+b^2\).
This means the quadratic can be rewritten as a square of a binomial.
The benefits of identifying a perfect square trinomial include simplification and ease of solving the quadratic equation.
In our example, the expression \(25x^2 + 70x + 49\) is a perfect square trinomial.
To see why, let's break it down:

• \((5x)^2=25x^2\)
• \(2 \times 5x \times 7=70x\)
• \(7^2=49\)

Since these match the structure \(a^2+2ab+b^2\), we can write this quadratic as \((5x+7)^2\). Recognizing perfect square trinomials enables quicker solving of equations.

Factoring Quadratics

Factoring quadratics involves breaking down a quadratic equation into the product of two binomials. This technique simplifies solving quadratics.
Factoring can help find solutions by making the equation easier to handle.
In the given equation \(25x^2 + 70x + 49=0\), recognizing it as a perfect square made factoring more straightforward.
We rewrote it as \((5x+7)^2=0\).
This factorization exposes the structure of the equation and is crucial for the next steps in solving.
Understanding how to factor helps to identify solutions efficiently and reveal potential insights about the equation's nature.

Solving Quadratic Equations

Once a quadratic equation is simplified, especially through factoring, solving it becomes straightforward.
When a quadratic is expressed as \((a+b)^2=0\), it implies that whatever is inside the square must equal zero.
In our scenario, the equation is \((5x+7)^2=0\).
Take the square root of both sides to simplify further:

• \(5x+7=0\)

By isolating the variable \(x\), we proceed with solving the equation.
The simplification from a perfect square trinomial directly assists in easily finding an answer by focusing on zeroing the inner expression.
These straightforward steps significantly simplify the often complex process of solving quadratics.

Real Solutions

Real solutions of a quadratic equation are the values of \(x\) that satisfy the equation. Not all quadratic equations have real solutions.
Some have complex solutions.
However, in our example, having a perfect square guarantee real solutions exist.
In the case of \((5x+7)^2=0\), the solution was found by isolating \(x\):

• Subtract 7: \(5x=-7\)
• Divide by 5: \(x=-\frac{7}{5}\)

This results in \(x=-\frac{7}{5}\) as the only real solution.
Understanding the nature of real solutions is critical, especially for determining the distinct or repeating nature of roots in an equation.
It's important to realize how the form of the quadratic impacts the type and number of solutions.