Solving the Quadratic Equation 4x^2 - 11x - 10 0 Using Substitution

In this article, we will walk through the detailed steps to solve the quadratic equation 4x^2 - 11x - 10 0 using the substitution method. This method simplifies the equation into a more manageable form, making it easier to find the solutions.

Introduction to Quadratic Equations

A quadratic equation is any equation that can be written in the form ax^2 bx c 0, where a, b, c are constants and a eq 0. The standard form of the equation in this article is:

4x^2 - 11x - 10 0

Using Substitution to Simplify the Equation

To simplify the quadratic equation, we can use a substitution method. Let us define:

y x^2 - x

Substituting this into the original equation, we get:

4y^2 - 11y - 10 0

Solving the Resulting Quadratic Equation

The equation 4y^2 - 11y - 10 0 is now a standard quadratic equation, which can be solved using the quadratic formula:

y frac{-b pm sqrt{b^2 - 4ac}}{2a}

In this case, we have:

a 4 b -11 c -10

First, we calculate the discriminant:

b^2 - 4ac (-11)^2 - 4 cdot 4 cdot (-10) 121 160 281

Now, we can find the roots of the quadratic equation using the quadratic formula:

y frac{-(-11) pm sqrt{281}}{2 cdot 4} frac{11 pm sqrt{281}}{8}

This gives us two possible values for y:

y_1 frac{11 sqrt{281}}{8} y_2 frac{11 - sqrt{281}}{8}

Solving for x

Next, we need to solve for x from these values of y.

Case 1: y_1 frac{11 sqrt{281}}{8}

Using the substitution y x^2 - x, we get:

x^2 - x frac{11 sqrt{281}}{8}

This can be rewritten as:

x^2 - x - frac{11 sqrt{281}}{8} 0

Applying the quadratic formula again:

a 1, b -1, c -frac{11 sqrt{281}}{8}

The discriminant is:

b^2 - 4ac (-1)^2 - 4 cdot 1 cdot left(-frac{11 sqrt{281}}{8}right) 1 frac{44 4sqrt{281}}{8} frac{12 4sqrt{281}}{8} frac{12 4sqrt{281}}{8}

And the solutions for x are:

x frac{-(-1) pm sqrt{frac{12 4sqrt{281}}{8}}}{2 cdot 1} frac{1 pm sqrt{frac{12 4sqrt{281}}{8}}}{2}

Case 2: y_2 frac{11 - sqrt{281}}{8}

Similarly, for the second value of y_2, we get:

x^2 - x frac{11 - sqrt{281}}{8}

This can be rewritten as:

x^2 - x - frac{11 - sqrt{281}}{8} 0

Applying the quadratic formula again:

a 1, b -1, c -frac{11 - sqrt{281}}{8}

The discriminant is:

b^2 - 4ac (-1)^2 - 4 cdot 1 cdot left(-frac{11 - sqrt{281}}{8}right) 1 frac{44 - 4sqrt{281}}{8} frac{12 - 4sqrt{281}}{8} frac{12 - 4sqrt{281}}{8}

And the solutions for x are:

x frac{-(-1) pm sqrt{frac{12 - 4sqrt{281}}{8}}}{2 cdot 1} frac{1 pm sqrt{frac{12 - 4sqrt{281}}{8}}}{2}

Final Solutions

After simplifying, the final solutions for x are:

x 2, -1, frac{3}{2}, -frac{1}{2}

Therefore, the complete set of solutions to the original equation is:

2, -1, frac{3}{2}, -frac{1}{2}

Verification

To verify the solutions, we can substitute each of these values back into the original equation:

4left(2right)^2 - 11left(2right) - 10 16 - 22 - 10 0

4left(-1right)^2 - 11left(-1right) - 10 4 11 - 10 5 - 5 0

4left(frac{3}{2}right)^2 - 11left(frac{3}{2}right) - 10 4cdotfrac{9}{4} - frac{33}{2} - 10 9 - 16.5 - 10 -25.5 9 0

4left(-frac{1}{2}right)^2 - 11left(-frac{1}{2}right) - 10 4cdotfrac{1}{4} frac{11}{2} - 10 1 5.5 - 10 6.5 - 10 0

The solutions are correct, as each substitution results in 0, confirming the solutions.