Solve quadratic equations by taking square roots and by completing the square, and use completing the square to write vertex form (LA A1: A-REI.B.4, A-SSE.B.3).

What this topic is asking

Standard A1: A-REI.B.4 also covers solving quadratics by taking square roots and by completing the square, and completing the square produces vertex form (linking to A-SSE.B.3). On LEAP 2025 these are Type I Major Content items. Square roots are best when there is no linear term; completing the square is the general method and the route to vertex form.

Solving by square roots

Use this when there is no linear term. Isolate the squared quantity, then take the square root of both sides with $\pm$.

A square equal to a negative number has no real solution, because no real number squares to a negative.

Completing the square

To make $x^2 + bx$ a perfect square, add $\left(\frac{b}{2}\right)^2$. The number inside the resulting binomial is half of $b$.

You must add the constant to both sides to keep the equation balanced.

Producing vertex form

Completing the square also rewrites $y = x^2 + bx + c$ in vertex form $y = (x - h)^2 + k$, where $(h, k)$ is the vertex. For $y = x^2 + 6x + 5$: $y = (x^2 + 6x + 9) + 5 - 9 = (x + 3)^2 - 4$, so the vertex is $(-3, -4)$. Vertex form is not on the reference sheet.

How LEAP examines this topic

• Equation response. Solve by square roots or completing the square and enter the solutions.
• Multiple choice. Identify the completing-the-square constant, or pick the vertex form.
• Drag and drop. Order the completing-the-square steps.

A clarifying idea: completing the square is the method that always works and is how the quadratic formula is derived, so understanding it explains where the formula comes from.

Why half of b, squared, completes the square

The constant $\left(\frac{b}{2}\right)^2$ completes the square because of how a binomial squares out, which is the algebraic fact behind A-REI.B.4 and A-SSE.B.3. Expanding $(x + p)^2$ gives $x^2 + 2px + p^2$: the middle coefficient is $2p$ and the constant is $p^2$. So if you have $x^2 + bx$ and want it to be a perfect square $(x + p)^2$, you need $2p = b$, which means $p = \frac{b}{2}$, and then the required constant is $p^2 = \left(\frac{b}{2}\right)^2$. That is exactly the rule: halve the linear coefficient and square it. This also explains why the number that ends up inside the binomial is $\frac{b}{2}$ (it is $p$), not $b$. Completing the square is powerful because it converts any quadratic into a squared term plus a constant, a form you can solve by a single square root and that displays the vertex directly. Crucially, applying it to the general equation $ax^2 + bx + c = 0$ and solving produces the quadratic formula, so completing the square is not just one method among several, it is the source of the universal solver, which is why mastering it deepens your grasp of every quadratic technique.

Try this

Q1. Solve $x^2 = 81$. [1 point]

• Cue. $x = \pm 9$.

Q2. What constant completes the square for $x^2 + 10x$, and what is the perfect square? [2 points]

• Cue. Half of $10$ is $5$, $5^2 = 25$; $(x + 5)^2$.