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Solve each equation. $$10 x-1=(2 x+1)^{2}$$

Short Answer

Expert verified
The solutions to the equation are \(x = 1\) and \(x = 0.5\).

Step by step solution

01

Expand the Square

Apply the formula \((a+b)^2 = a^2 + 2ab + b^2\) to \( (2x+1)^2 \) to get \( 4x^2 + 4x + 1 \). The equation now is \( 10x - 1 = 4x^2 + 4x + 1 \).
02

Rearrange the Equation

Rearrange the equation to the standard quadratic form \(ax^2 + bx + c = 0\). This results in \( 4x^2 - 6x + 2 = 0\) after subtracting \(10x\) and adding \(1\) to both sides.
03

Apply the Quadratic Formula

Use the quadratic formula to solve for \(x\), which is \( x = [-b \pm \sqrt{b^2 - 4ac}]/(2a)\). Plugging \( a = 4\), \( b = -6 \), and \( c = 2\) into the formula gives solutions \( x = 1\) and \( x = 0.5\).

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