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A quadratic equation is a second-degree polynomial equation in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The highest power of the variable \( x \) is 2 in a quadratic equation. These equations are foundational in algebra and appear in various scientific and engineering problems.
To solve a quadratic equation, one common method is factoring, which involves expressing the quadratic as a product of two binomials. By factoring, you can find the roots, or solutions, of the equation.
The general idea is to convert the quadratic equation into a form such as \((dx + e)(fx + g) = 0\). This makes it easier to find the values of \( x \) that satisfy the equation. If the quadratic can be factored, then for each factor, the solutions can be derived by setting the factor equal to zero.

A common factor is a number or algebraic expression that divides all terms of a given expression without leaving a remainder. Finding the common factor is a vital initial step in factoring any polynomial, including quadratic equations.
In the equation \(75x^2 + 35x - 10\), we look for a common factor among the coefficients 75, 35, and -10. In this case, 5 is a common factor because each coefficient is divisible by 5. Factoring out the common element simplifies the expression:

• Divide each term by 5: \( 75x^2 \rightarrow 15x^2 \)
• \(35x \rightarrow 7x\)
• \(-10 \rightarrow -2\)

This simplification helps reduce the equation to \(5(15x^2 + 7x - 2) = 0\), making it easier to apply further factoring techniques.

Factoring by grouping is a strategy used to factor certain types of polynomials by grouping terms with common factors and then factoring out these common terms.
In the simplified quadratic equation \(15x^2 + 7x - 2\), it can be challenging to factor directly. Here, we use a technique involving the product and sum of numbers.
First, identify two numbers that multiply to \(15 \times -2 = -30\) and add to 7. These numbers are 10 and -3. We use these to rewrite the middle term, resulting in the equation \(15x^2 + 10x - 3x - 2\).
The next step is grouping the terms as \((15x^2 + 10x) + (-3x - 2)\). Factor out the greatest common factor from each group:

• \(15x^2 + 10x\) gives us \(5x(3x + 2)\)
• \(-3x - 2\) simplifies to \(-1(3x + 2)\)

Finally, since \(3x + 2\) is common in both expressions, we can factor it out as a binomial: \((5x - 1)(3x + 2) = 0\).
Once factored, solving the equation involves finding the values of \( x \) that satisfy \((5x - 1) = 0 \) or \((3x + 2) = 0 \). This method, known as factoring by grouping, offers a way to handle trinomials that cannot be factored directly by simple methods.