Reverse an expansion to make a square
Start with (x − 2)² = (x − 2)(x − 2). Multiplying the terms gives x² − 2x − 2x + 4 = x² − 4x + 4. Keeping only x² and 4 would lose both cross terms. To check a squared bracket, return to the product of its two identical factors.
In x² − 4x + 7, split the constant 7 into 4 + 3. The part x² − 4x + 4 is (x − 2)², so the original expression is (x − 2)² + 3. You have used four of the original seven to form the square and left three outside; you have not increased the expression by four.
In general, x² + bx + c = (x + b/2)² + c − b²/4. Expanding the bracket gives x² + bx + b²/4, and the outside term −b²/4 removes the extra constant. Expanding back to the original three terms checks the sign and constant before you rely on the formula.
Show a lower bound, then show that it is attained
The square of any real number is at least zero. Thus (x − 2)² + 3 is at least 3 for every real x. At x = 2 the square is zero, so the output really is 3. Both facts are needed: no output is below 3, and an allowed input produces 3.
The first exercise asks for the value of the expression, rather than the value of x. Two is the minimizing input; three is the minimum output. The original constant 7 is the output at x = 0, but it is not the minimum. Substituting x = 2 into the original expression also gives 4 − 8 + 7 = 3.
Read the input range as well. Under the different practice condition 3 ≤ x ≤ 4, x = 2 is unavailable. Now x − 2 lies between 1 and 2, so its square is at least 1. This bound is attained at x = 3, making the minimum of the same expression 4 on this interval. The actual exercise permits all real inputs, so it permits the input 2.
Read symmetry from equally distant inputs
For y = (x − 2)² + 1, the inputs x = 1 and x = 3 both give y = 2. They are equally far to the left and right of 2. More generally, x = 2 − t and x = 2 + t put −t and t inside the square, giving equal squared values.
This left-right pairing is symmetry about the vertical line x = 2. The lowest point on this graph is called its vertex, which here is (2, 1). The symmetry axis is neither the horizontal line at the vertex's height, y = 1, nor the horizontal line y = 2. On the axis the horizontal coordinate stays 2, so its equation is x = 2.
The expression x − 2 does not place the axis at −2. It becomes zero when x = 2. In y = (x − h)² + k, the inputs h − t and h + t likewise give equal heights: the axis is x = h and the vertex is (h, k). Distinguish the minimum height k from the horizontal coordinate h where it is attained.
Count real solutions by asking whether zero is reachable
The final exercise, x² + 2x + 5 = 0, becomes (x + 1)² + 4 = 0. Its left side is at least 4 for every real input, so it never reaches zero. There are therefore zero distinct real solutions. Being a quadratic equation does not by itself guarantee two real solutions.
Compare three original practice equations that change the constant. The equation (x + 1)² + 4 = 0 has no real solution. The equation (x + 1)² = 0 has the single solution x = −1. For (x + 1)² − 4 = 0, x + 1 is 2 or −2, giving x = 1 or −3. Ask whether the square is required to be negative, zero or positive.
The Japanese exercise explanation uses the discriminant D, calculating D = 2² − 4 × 1 × 5 = −16. For x² + bx + c, the constant left outside the completed square is c − b²/4 = −D/4, where D = b² − 4c. Here 4 = −(−16)/4. A positive four that prevents the expression from reaching zero and a negative discriminant describe the same condition in two forms.
Return the three exercises to the same form
The exercises below ask for a minimum, a symmetry axis and a real-solution count, in that order. Before opening the answers, write the square-plus-constant form and name the quantity requested. After reading an explanation, hide it and reconstruct the expansion, the minimizing input, the equally distant inputs and the condition for reaching zero.
If the roles of letters or equality are unclear, return to the prerequisite article “Expressions: give the unknown a role.” The related article “Graphs and coordinates: read what each axis means” helps distinguish horizontal and vertical coordinates. The practice preset selects high-school-level, standard questions on expressions and functions. It includes other matching questions, rather than promising only these three examples.